Hello fellow robotic enthusiasts! 
Let’s dive into the intriguing world of defining reference frames in robotics. There are multiple methods out there, but today, I want to shed some light on the 3-2-1 method, a personal favorite of mine.
The 3-2-1 Method:
In this method, we use a combination of points to establish a reference frame:
- 3 points define the top plane.
- 2 points contribute to an adjacent plane.
- And finally, 1 point locks in the last plane.
It’s like connecting the dots in space! 
This method offers a structured and intuitive way to set up a reference frame, crucial for accurate robotic movements.
In that scenario, here’s a code example that will output a reference frame in multiple formats.
# Here's where you apply the TCP you will use during probing
is_successful = robot.set_tcp_offset(toolingOnly_offset)
print("TCP for Calibration: ",is_successful)
time.sleep(5)
### Input the 6 probed points here ###
pointList = np.array([
[255.63,-601.24,-49.20,-100.99,87.19,-170.23],# A
[-147.10,-746.40,-43.68,89.46,60.19,-0.67],# A
[-204.32,-523.36,-50.08,-92.06,-53.36,179.83],# A
[350.15,-491.66,-72.55,-92.81,-0.40,-89.82],# B
[352.25,-697.07,-64.41,99.11,1.33,-89.47],# B
[196.74,-378.35,-61.60,178.99,-2.10,179.75] # C
])
### Reference Frame Calibration Section ###
def getReferenceFrame(points):
# plan A
pt1 = points[0,:3]
pt2 = points[1,:3]
pt3 = points[2,:3]
v21 = pt2-pt1
v23 = pt2-pt3
NormalA =np.cross(v21,v23)
NormalA = NormalA / np.linalg.norm(NormalA)
Da = np.dot(NormalA,pt1)
print(f"plan A equation: {NormalA[0]:.2f}x + {NormalA[1]:.2f}y + {NormalA[2]:.2f}z = {Da}")
# plan B
pt4 = points[3,:3]
pt5 = points[4,:3]
ptb = pt4 + NormalA
v54 = pt5-pt4
v5b = pt5-ptb
NormalB =np.cross(v54,v5b)
NormalB = NormalB / np.linalg.norm(NormalB)
Db = np.dot(NormalB,pt4)
print(f"plan B equation: {NormalB[0]:.2f}x + {NormalB[1]:.2f}y + {NormalB[2]:.2f}z = {Db}")
#plan C
pt6 = points[5,:3]
ptc1 = pt6 + NormalA
ptc2 = pt6 + NormalB
v6c1 = pt6 - ptc1
v6c2 = pt6 - ptc2
NormalC =np.cross(v6c1,v6c2)
NormalC = NormalC / np.linalg.norm(NormalC)
Dc = np.dot(NormalC,pt6)
print(f"plan C equation: {NormalC[0]:.2f}x + {NormalC[1]:.2f}y + {NormalC[2]:.2f}z = {Dc}")
# find interesction of the 3 planes (solve Ax = b)
A = np.array([NormalA, NormalB, NormalC])
b = np.array([Da, Db, Dc])
# Solve the system of linear equations
try:
origin = np.linalg.solve(A, b)
except np.linalg.LinAlgError:
# The planes are parallel or coincident, no unique solution
return None
# Z axis, along NormalA, check if pointing up or down with pt4
def adjustOrientation(Normal,pt):
dir = pt - origin
upOrDown = np.dot(dir,Normal)
if upOrDown < 0:
return -Normal
else:
return Normal
zAxis = -adjustOrientation(NormalA,pt6) # z axis is inverted by convention
xAxis = adjustOrientation(NormalB,pt6)
yAxis = adjustOrientation(NormalC,pt1)
print("normalA (Z)",NormalA,zAxis)
print("normalB (X)",NormalB,xAxis)
print("normalC (Y)",NormalC,yAxis)
rot = np.zeros( (3,3))
rot[:,0] = xAxis
rot[:,1] = yAxis
rot[:,2] = zAxis
r = scipyR.from_matrix(rot)
print("Rotation Matrix:",rot)
eulr = r.as_euler('xyz', degrees=True)
return [origin[0],origin[1],origin[2],eulr[0],eulr[1],eulr[2]]
result = getReferenceFrame(pointList)
print(result)
Why I Like It:
I find the 3-2-1 method provides a nice balance between simplicity and accuracy. It’s especially handy when working on tasks where a well-defined reference frame is essential for success.
Your Thoughts?
Have you explored different methods for defining reference frames? What’s your go-to approach? Share your experiences or any tips you might have! Let’s exchange ideas and knowledge. 
