Shoulder Rehab: Resistance-Band ROM Simulator

Resistance Band Level

Exercise Setup

Target ROM (raise to) 150°

Rep cadence 12 rpm

Patient strength 14 N·m

Live Readouts

Joint angle

0.0°

Band tension

0.0 N

Reps completed

Muscle torque

0.0 N·m

Movement quality 100%

How it works

The arm is modelled as a rigid pendulum of length L pivoting at the shoulder. Newton's second law for rotation gives the joint's angular acceleration:

I·α = τ_muscle − τ_gravity − τ_band − c·ω
τ_gravity = m·g·L·sin(θ)
τ_band    = k·(‖band‖ − L₀) · (moment arm)   [Hooke's law, only while stretched]
τ_muscle  = clamp(Kp·(θ_target − θ) − Kd·ω, −τ_max, +τ_max)

θ is the shoulder-abduction angle (0° = arm at side, 180° = arm overhead). The target angle θ_target sweeps smoothly between 0° and the prescribed ROM at the chosen cadence, and a PD controller stands in for the patient's active muscle effort — capped at the patient-strength limit τ_max, exactly the way real strength loss limits how far or how fast someone can move against resistance.

Band level — sets the elastic stiffness k (N/m); a stiffer band means more resisting torque for the same stretch.
Target ROM — the prescribed range of motion for the rep.
Cadence — reps per minute; faster reps raise the required angular velocity and thus the damping/inertial load.
Patient strength — the torque ceiling τ_max; when it can't overcome gravity + band torque at the target angle, the arm visibly falls short of full ROM — the same "can't complete the rep" pattern a therapist watches for.
Movement quality — 100% minus a penalty built from tracking error (how far actual θ lags θ_target) and jerk (rate of change of angular acceleration), the two hallmarks clinicians use to judge whether a rep was smooth and controlled versus jerky or incomplete.

Real-world relevance: this torque-balance + PD-tracking model is the same structure used in robotic rehab devices and biomechanics research to quantify how resistance, cadence and residual strength interact during progressive-loading therapy.

Actual arm position Target trajectory Resistance band

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Shoulder Rehabilitation: Resistance-Band ROM Simulator