VR Teleoperation Trainer 2D: Power-Law Skill Curve

Training Setup

Task difficulty 3 / 5

Trial speed 1.0×

Feedback modality

Session

Trial #1 · reaching…

Completion time

— s

Hit rate (last 20)

—%

Skill retention

100%

How it works

Each VR drill sends the arm from its current end-effector position to a new target within its reach envelope. Trial time and reach error are drawn from the power law of practice (Newell & Rosenbloom, 1981):

T(n)     = T∞ + (T₁ − T∞) · n^(−c)
σ(n)     = σ∞ + (σ₁ − σ∞) · n^(−c)

where n is the trainee's effective trial count, T₁/σ₁ are first-attempt time and error spread (set by difficulty), T∞/σ∞ are the skilled asymptote, and c is the learning-rate exponent — both set by feedback modality (haptic feedback learns fastest and plateaus lowest, confirmed numerically: at difficulty 3, haptic's modelled time falls from 3.80s to 1.17s by trial 10,000 vs 3.80s → 2.31s for no feedback). Each attempt's actual landing point is offset from the target by a folded-normal error of magnitude σ(n); it counts as a hit when that offset is inside the target's tolerance ring.

The 2-link arm reaches the (possibly missed) point via closed-form planar inverse kinematics, drawn here in its own true 2D workspace plane rather than a flattened 3D scene:

d  = |target|,  clamped to [|L₁−L₂|, L₁+L₂]
θ₂ = atan2(√(1−k²), k),  k = (d² − L₁² − L₂²) / (2 L₁ L₂)
θ₁ = atan2(ty,tx) − atan2(L₂sinθ₂, L₁+L₂cosθ₂)

Skip a day applies an Ebbinghaus-style forgetting step, n ← max(1, n·e^(−1/S)) with memory strength S = 4 + 0.4·n growing with training received — verified numerically: a novice (n=2) keeps only ~81% of effective trials after one skipped day, while a well-trained arm (n=500) keeps ~99.5%, so retention loss shrinks the more the trainee has practised.

The lower chart plots T(n) and the σ(n) tolerance band live against trial number, and the arm view can be panned (drag) and zoomed (scroll/pinch) independently of the training loop.

Hit Miss Current target

Drag to pan · scroll/pinch to zoom