I’ve been playing around with the idea of building a rig to suspend an object, like a camera or prop, using three wires and then control its position in 3-D space using a few stepper motors and a micro-controller. The general idea is similar to the skycam system they use at football games and other pro sports, or a large scale delta robot. The first thing I needed to figure out was how much I wanted this rig to be able to hold and how much tension (force) would be required to hold and move this weight around. Additional I wanted to be able to visualize the results so I could look for any peaks, valleys or other anomalies that would throw off my calculations. I ended up writing the python script below utilizing the mpmath, sympy and matplotlib libraries to calculate the tension on a wire due to a suspend mass at a specific height and plot it as the weight moves around the x/y plane. The image below is a sample of the output from this script for a 5 kg mass suspend 2.0 m above the ground with three anchor points at 3.0 m above ground level and each 120 degrees apart.

from mpmath import *
from sympy import *

# Gravity constant
g = 9.81

# Length function
def length(m):
    s = 0.0
    for n in range(len(m)):
        s += m[n]**2

    return sqrt(s)

# Tension Function
def tension(x, y, z):
    # Location of object
    loc = Matrix([[x, y, z]])

    # Calculate x, y, z length of each wire
    la = pta - loc
    lb = ptb - loc
    lc = ptc - loc

    # Calculate the tension ratio of each wire
    ra = la / length(la)
    rb = lb / length(lb)
    rc = lc / length(lc)

    # Put ratios into matrix form        
    a = Matrix([ra, rb, rc])

    # Calculate solution
    sol = lu_solve(a.T, f)

    # Check if results are valid
    for n in range(len(sol)):
        if (sol[n] < 0):
            sol[:] = 'nan'

    # Return tension for wire 'a'
    return sol[0]

# Locations of anchors points [[ x, y, z]]
pta = Matrix([[ 0.00,  2.00, 3.0]])
ptb = Matrix([[-1.73, -1.00, 3.0]])
ptc = Matrix([[ 1.73, -1.00, 3.0]])

# Enter the mass of the object suspended
mass = input("Enter mass of object (Kg): ")

# Enter height of object
z = input("Enter height of object (m): ")

# Force due to gravity on object [x, y, z]
f = Matrix([0.0, 0.0, mass * g])

# Create a symbolic function to return tension at position
force = lambda x, y: tension(x, y, z)

# Plot tension on wire over area of anchor points
splot(force, [-1.73, 1.73], [-1.0, 2.0])

 Change Log:

1/4/2014: Added check to tension function to filter out invalid results. This makes the edges of the plot a little rough but gives a better visual.