I have built a simple robotic arm using 3 RC Servos and an Arduino. I just want to play around with it and learn something about robotics.
Currently, I am trying to compute the position of the tip of the robotic arm using the three angular positions of the servos. "Forward kinematics" I think is the technical term for this. Btw the tip of the arm is a pen, I thought I might try to draw something with it later on.
In the movement range of the arm I set up a Cartesian coordinate system and recorded 24 (angle => position) samples. pastebin.com/ESqWzJJB
Now, I am trying to model this data, but I am a bit out of my depth here. Here is my approach so far:
I use the Denavit–Hartenberg equations found on Wikipedia en.wikipedia.org/wiki/Denavit–Hartenberg_parameters. I then try to determine the parameters using least squares optimization.
minimize(sum(norm(f(x,P)-y)^2))
I also added linear terms to the input and output of the model to compensate for possible distortions (e.g. phase-shift in the servo angle):
y = f(ax+b)*c+d
My Python code: pastebin.com/gQF72mQn
from numpy import *
from scipy.optimize import minimize
# Denavit-Hartenberg Matrix as found on Wikipedia "Denavit-Hartenberg parameters"
def DenHarMat(theta, alpha, a, d):
cos_theta = cos(theta)
sin_theta = sin(theta)
cos_alpha = cos(alpha)
sin_alpha = sin(alpha)
return array([
[cos_theta, -sin_theta*cos_alpha, sin_theta*sin_alpha, a*cos_theta],
[sin_theta, cos_theta*cos_alpha, -cos_theta*sin_alpha, a*sin_theta],
[0, sin_alpha, cos_alpha, d],
[0, 0, 0, 1],
])
def model_function(parameters, x):
# split parameter vector
scale_input, parameters = split(parameters,[3])
translate_input, parameters = split(parameters,[3])
scale_output, parameters = split(parameters,[3])
translate_output, parameters = split(parameters,[3])
p_T1, parameters = split(parameters,[3])
p_T2, parameters = split(parameters,[3])
p_T3, parameters = split(parameters,[3])
# compute linear input distortions
theta = x * scale_input + translate_input
# load Denavit-Hartenberg Matricies
T1 = DenHarMat(theta[0], p_T1[0], p_T1[1], p_T1[2])
T2 = DenHarMat(theta[1], p_T2[0], p_T2[1], p_T2[2])
T3 = DenHarMat(theta[2], p_T3[0], p_T3[1], p_T3[2])
# compute joint transformations
# y = T1 * T2 * T3 * [0 0 0 1]
y = dot(T1,dot(T2,dot(T3,array([0,0,0,1]))))
# compute linear output distortions
return y[0:3] * scale_output + translate_output
# least squares cost function
def cost_function(parameters, X, Y):
return sum(sum(square(model_function(parameters, X[i]) - Y[i])) for i in range(X.shape[0])) / X.shape[0]
# ========== main script start ===========
# load data
data = genfromtxt('data.txt', delimiter=',', dtype='float32')
X = data[:,0:3]
Y = data[:,3:6]
cost = 9999999
#try:
# parameters = genfromtxt('parameters.txt', delimiter=',', dtype='float32')
# cost = cost_function(parameters, X, Y)
#except IOError:
# pass
# random init
for i in range(100):
tmpParams = (random.rand(7*3)*2-1)*8
tmpCost = cost_function(tmpParams, X, Y)
if tmpCost < cost:
cost = tmpCost
parameters = tmpParams
print('Random Cost: ' + str(cost))
savetxt('parameters.txt', parameters, delimiter=',')
# optimization
continueOptimization = True
while continueOptimization:
res = minimize(cost_function, parameters, args=(X,Y), method='nelder-mead', options={'maxiter':100,'xtol': 1e-5})
parameters = res.x
print(res.fun)
savetxt('parameters.txt', parameters, delimiter=',')
continueOptimization = not res.success
print(res)
But it just won't work, none of my attempts have converged on a good solution. I also tried a simple 3x4 matrix multiplication, which does not make much sense as a model, but oddly it didn't do worse than the more sophisticated model above.
I hope there is someone out there who can help.
