This is an independent 2D top-down model of the same ISO/TS 15066 Power and Force Limiting (PFL) safety budget as the 3D version, computed from real forward kinematics and a live speed-and-separation-monitoring (SSM) controller instead of a single scripted "run test" animation — everything below recomputes every animation frame from the current arm pose and hand position.
Forward kinematics (2-link planar arm, angles θ1, θ2):
ex = L1·cos θ1 + L2·cos(θ1+θ2)
ey = L1·sin θ1 + L2·sin(θ1+θ2)
v_R = |Δ(ex,ey)/Δt| real numeric end-effector speed
Minimum protective separation distance (ISO 13855 / 15066 SSM):
Ss = v_R·Tr + v_R²/(2·a_max) robot stopping distance
Sh = v_H·(Tr + Ts), Ts = v_R/a_max human approach during robot's stop
S_min = Ss + Sh + C C = fixed intrusion allowance
Predicted transient contact force (impulse form of the PFL contact model):
μ = (m_R·m_H)/(m_R+m_H) reduced mass
F = μ·v_closing / Ts μ decelerated to 0 within stopping time Ts
Every frame, the controller compares the arm's actual separation S from the dragged hand against the S_min the current joint speed would require to stop in time. If S drops below S_min the speed governor throttles the commanded joint speed down (down to a full stop at the intrusion limit C), which is what visibly slows or halts the arm — the slowdown is a consequence of the formula, not a scripted cue. The force bar meanwhile shows what a transient contact would produce right now, at the arm's actual closing speed and the selected body region's reduced mass, against that region's published ISO/TS 15066 Annex A force limit.
- Region — sets the body's effective mass mH and its transient contact force limit.
- ωmax — the task's nominal (unthrottled) joint speed; the governor scales it down live.
- mR — the arm's reflected mass at the end-effector, used in the reduced-mass μ.
- amax / Tr — the safety controller's real braking capability and sensor/control reaction latency, both of which lengthen S_min as they get worse.
Real-world relevance: this is exactly the two safety layers a certified cobot deployment must both pass — SSM keeps the arm far enough from a person that it can always stop in time, and PFL bounds the force of any contact that still happens despite that margin.