Robotics · Construction (façade retrofit)
Eight Lines
A design model of an eight-cable robot for façade panels: where it can hold the panel, and where the wind would move it more than 5 mm.
Overview
A design model of a cable-driven robot that installs façade panels, in simulation. Eight steel cables run from roof outriggers, parapet arms, ground plates and two street winch frames to a carrier holding a 2.4 × 1.2 m panel (270 kg with the carrier) in front of a fictional six-storey façade, 24 m wide. For every position of the panel, the model solves the eight cable tensions. It maps where all eight can hold the panel, taut and within their limits, and where wind would push it more than ±5 mm from its target.
In still air the cables can hold the panel at 83.8% of the positions on the façade; the top corners and the lowest row of panels are out of reach. In a 9 m/s wind (10-minute mean at 10 m height) with its peak gusts, they can still hold it at 76.4%. But with the winches locked, the panel stays within ±5 mm at only 21.2%. At 8 m/s that figure is 54.5%, and at 10 m/s it is zero. The weak direction is out of the façade plane. Every cable has to stay clear of the panels already installed, so the cables that pull the carrier towards the wall run almost parallel to it.
Moving the two street winch frames further out raises the area within ±5 mm at 8 m/s from 54.5% to 64.6%, at the cost of reach in still air (83.8% → 80.7%). Moving the street frames and the roof outriggers closer in (5 m and 2 m out) gives the most reach in still air (86.8%) and the least accuracy (15.9% at 8 m/s). One gust runs in a physics simulation (MuJoCo) at an upper bay, 16.8 m up. A 1-second gust up to the peak gust speed of the 9 m/s wind moves the worst panel corner 7.3 mm, 6% more than the static model's 6.9 mm. A sharper 0.5-second gust moves it 9.5 mm, and the panel takes 4.6 s after the gust to settle back within ±0.5 mm.
Scope: a quasi-static design model. The panel stays parallel to the façade. Cables are steel ropes with self-weight sag (parabolic approximation), not simulated cable dynamics. Wind loads follow a stated EN 1991-1-4-style assumption, not a code check. During a gust the winches hold their lengths, with no feedback correction.
What we built
- Tension distribution for eight cables and six degrees of freedom. For each pose, the model finds the exact set of feasible tensions (a polygon in the two-dimensional null space) and, within it, the tensions closest to mid-range, after Gouttefarde et al. (IEEE T-RO, 2015). In the browser it takes about 0.03 ms per solve.
- Wrench-feasible workspace maps. The required load is gravity plus four wind cases: pressure towards and away from the wall, each with the ±0.25 b eccentricity used for signboards. Feasibility is checked with a hyperplane-shifting capacity margin. Each rope's lower tension limit keeps its sag under 1% of its length.
- Stiffness and placement accuracy. Each rope's stiffness is reduced for sag (Ernst's equivalent modulus), and the tension (geometric) terms are added. With the winches locked, the full nonlinear equilibrium under each wind case gives the worst movement of any panel corner, which is checked against a ±5 mm gate.
- A MuJoCo gust case. The panel and carrier are a free body on eight tendons that can only pull. The tendons use the same sagging-rope law and add damping, and the wind force uses the panel's relative velocity.
- An interactive model. Drag the panel and the eight tensions solve live. You can change the wind, switch between the reach and ±5 mm maps, drag the street frames, or let a search move them. All of it uses the same equations as the films, ported to JavaScript and checked against the Python model.
Tech stack
Python · NumPy · SciPy (HiGHS) · MuJoCo · JavaScript (ES modules, Web Workers) · three.js
Numbers
| Number | How we measured it |
|---|---|
| 83.8% | Share of panel positions on the 24 × 21 m façade where tensions within limits exist in still air. Each tension stays between t_min (1 kN, or the tension that keeps the rope's sag ≤ 1% of its length) and 25 kN. Checked on a 0.3 m grid of 4,745 panel-centre positions. |
| 76.4% / 21.2% | At a 9 m/s wind: the share of positions where the cables can hold the panel against gravity plus all four wind cases, and the share where, with the winches locked, no panel corner moves more than 5 mm and no rope drops below 1 kN or rises above 25 kN. Wind pressure is the EN 1991-1-4-style peak velocity pressure at panel height (terrain category III), with a force coefficient of 1.8 on the panel. Same grid. |
| 54.5% → 0% | Share of positions within ±5 mm at 8 m/s and at 10 m/s, same method. The median corner movement goes from 4.5 mm to 7.4 mm: a little faster than the square of the wind speed, because ropes soften as they unload. |
| 54.5% → 64.6% | Share within ±5 mm at 8 m/s, before and after moving the two street winch frames from 9 m to 14 m out from the façade, from 16 m to 18 m either side of its centre line, and from 1.0 m to 0.5 m high. The search was a grid plus pattern search on a 1.2 m grid; the result was re-evaluated on the 0.3 m grid. Reach in still air falls from 83.8% to 80.7%. |
| 6.9 mm → 7.3 mm | Worst panel-corner movement at the bay 16.8 m up, in the 9 m/s wind: static model vs. MuJoCo with a 1-second 1-cos gust from still air to the peak gust speed (17.1 m/s at panel height). 7.1 mm of the dynamic movement is out of the façade plane. |
| 9.5 mm, 4.6 s | The same bay and wind with a sharper 0.5-second gust: the dynamic peak (+38% on the static model), and the time after the gust until the panel stays within ±0.5 mm. Damping is 2% in the lowest mode (assumed), plus aerodynamic damping from the panel's own motion. |
| 0.36 vs 2.1–2.5 kN/mm | Translational stiffness of the carrier at the bay 16.8 m up: out of the façade plane, against the two in-plane directions. Diagonal terms of the 6 × 6 stiffness matrix at the still-air holding tensions (rope stretch, sag and tension terms). |
| 4.70 Hz | Out-of-plane sway frequency of the panel in MuJoCo after the gust (4.68–4.70 Hz across the four gust lengths we ran). The stiffness matrix of the static model predicts 4.70 Hz for that mode. |
| 1,998 of 1,998 | Random pose and wind cases where our tension solver and an independent linear program (HiGHS) agree on whether tensions within limits exist. |
| 0.03 ms | Time per eight-cable tension solve in the browser (desktop Chrome on an Apple M1 Pro laptop). |
Stills
Try it live
An interactive view that runs in your browser. Nothing loads until you open it.
