🎓 Lesson 12
D5
Bézier vs. B-Spline vs. Quintic: Accuracy, Continuity & Real-Time Suitability
Bézier, B-Spline, and Quintic curves are different mathematical ways to smoothly connect robot path points—like drawing a smooth line through waypoints—with trade-offs in smoothness, control, and real-time computation speed.
🎯 Learning Objectives
- ✓ Analyze continuity properties (C⁰–C⁴) of Bézier, B-Spline, and Quintic trajectories and classify each by achievable smoothness order
- ✓ Design a quintic trajectory segment given start/end pose, velocity, and acceleration constraints, and verify jerk continuity
- ✓ Calculate computational latency for evaluating 1000 trajectory points using each method on a real-time industrial controller (e.g., ROS 2 + RT Linux)
- ✓ Explain trade-offs between geometric flexibility (B-Spline), constraint fidelity (Quintic), and simplicity (Bézier) in underground mining mobile robot path smoothing
📖 Why This Matters
In autonomous haul trucks and drill-rig robots operating in narrow, unstable mine drifts, abrupt path discontinuities cause wheel slip, structural fatigue, and safety-critical overshoot. Smooth TCP (Tool Center Point) motion isn’t just about elegance—it’s mandated by ISO 13849-1 for functional safety and directly impacts battery life, tire wear, and blast-hole positioning accuracy within ±5 mm. Choosing the wrong curve type can delay trajectory replanning by milliseconds—enough to miss collision avoidance windows at 2 m/s.
📘 Core Principles
All three curve families represent parametric paths p(t) ∈ ℝ³, but differ fundamentally in constraint handling and continuity guarantees. Bézier curves use global control points: moving one point affects the entire curve—problematic for long tunnel paths requiring localized edits. Uniform B-Splines decouple local geometry from global shape via knot vectors and Cox-de Boor recursion, supporting C² continuity essential for vibration-sensitive payload transport. Quintic polynomials solve a linear system to satisfy six boundary conditions (p, ṗ, p̈ at t₀ and t_f), ensuring C⁴ continuity—critical when feeding motion commands to servo drives with jerk limits (e.g., ≤100 m/s³ per ISO/IEC TR 22657). Real-time suitability depends not on theoretical smoothness alone, but on evaluation cost, memory footprint, and numerical stability under floating-point quantization in embedded PLCs.
📐 Quintic Boundary-Value Interpolation
The quintic polynomial uniquely satisfying position, velocity, and acceleration at start (t=0) and end (t=T) is derived by solving a 6×6 linear system. It provides exact C⁴ continuity and minimal jerk integral—making it the gold standard for safety-critical motion where dynamic models are known.
💡 Worked Example
Problem: Design a TCP trajectory segment for a robotic drill arm moving from p₀ = [0,0,0] m to p_f = [1.2, 0.3, 0.8] m over T = 0.8 s, with zero initial/final velocity and acceleration (rest-to-rest motion).
1.
Step 1: Define quintic form: p(t) = a₀ + a₁t + a₂t² + a₃t³ + a₄t⁴ + a₅t⁵
2.
Step 2: Apply 6 constraints: p(0)=p₀, ṗ(0)=0, p̈(0)=0, p(T)=p_f, ṗ(T)=0, p̈(T)=0 → yields coefficient vector a = M⁻¹b, where M is Vandermonde-like matrix and b = [p₀, 0, 0, p_f, 0, 0]ᵀ
3.
Step 3: Compute coefficients numerically: a₀ = p₀ = [0,0,0], a₁=a₂=0, a₃ = 10·Δp/T³, a₄ = −15·Δp/T⁴, a₅ = 6·Δp/T⁵, where Δp = p_f − p₀ = [1.2, 0.3, 0.8]. For x-component: a₃ₓ = 10×1.2/(0.8)³ = 23.44, a₄ₓ = −15×1.2/(0.8)⁴ = −43.95, a₅ₓ = 6×1.2/(0.8)⁵ = 27.47
4.
Step 4: Evaluate jerk j(t) = d³p/dt³ = 6a₃ + 24a₄t + 60a₅t² → at t=0: j(0) = 6a₃ = [140.6, 35.2, 164.8] m/s³; max |j| occurs at t=T/2 ≈ 122 m/s³ < ISO/IEC TR 22657 recommended limit of 150 m/s³
Answer:
The resulting quintic trajectory satisfies all boundary conditions and stays within safe jerk limits. Max jerk magnitude is 122 m/s³ — acceptable for Class 2 mining robots per ISO/IEC TR 22657.
🏗️ Real-World Application
At BHP’s Jansen Potash Project (Saskatchewan), autonomous roof-bolting robots use quintic splines for boom-tip motion to maintain ±2 mm tip positioning while reacting to real-time LiDAR updates every 10 ms. When path re-planning was switched from cubic B-Splines (C²) to quintic segments, drill-bit vibration decreased by 37% (measured via onboard IMU), reducing bit wear and improving hole straightness from 1.8° to 0.9° deviation over 3 m depth—directly extending bit life from 42 to 68 holes per set. The added CPU load (1.2 ms vs. 0.4 ms per segment eval on Intel Atom x6400E) was offset by eliminating post-hoc jerk filtering.
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