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IK_debug.py
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IK_debug.py
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# -*- coding: utf-8 -*-
from sympy import *
from time import time
from mpmath import radians
import tf
'''
Format of test case is [ [[EE position],[EE orientation as quaternions]],[WC location],[joint angles]]
You can generate additional test cases by setting up your kuka project and running `$ roslaunch kuka_arm forward_kinematics.launch`
From here you can adjust the joint angles to find thetas, use the gripper to extract positions and orientation (in quaternion xyzw) and lastly use link 5
to find the position of the wrist center. These newly generated test cases can be added to the test_cases dictionary.
'''
test_cases = {1:[[[2.16135,-1.42635,1.55109],
[0.708611,0.186356,-0.157931,0.661967]],
[1.89451,-1.44302,1.69366],
[-0.65,0.45,-0.36,0.95,0.79,0.49]],
2:[[[-0.56754,0.93663,3.0038],
[0.62073, 0.48318,0.38759,0.480629]],
[-0.638,0.64198,2.9988],
[-0.79,-0.11,-2.33,1.94,1.14,-3.68]],
3:[[[-1.3863,0.02074,0.90986],
[0.01735,-0.2179,0.9025,0.371016]],
[-1.1669,-0.17989,0.85137],
[-2.99,-0.12,0.94,4.06,1.29,-4.12]],
4:[[[2.15286,0,1.94653],
[0.0,-0.000148353,0.0,1.0]],
[1.84986,0.0,1.94645],
[0.00,0.00,0.00,0.00,0.00,0.00]],
5:[[[-1.07403,0.303366,0.649053],
[-0.471182,-0.05605,0.790594,0.38705]],
[-0.99635,0.101926,0.86165],
[3.04,-0.32,1.08,5.13,0.77,-5.02]],
}
# Make Homegeneous Matrix from DH parameters
def matrix_from(alpha, a, q, d):
return Matrix([[ cos(q), -sin(q), 0, a ],
[ sin(q)*cos(alpha), cos(q)*cos(alpha), -sin(alpha), -sin(alpha) * d],
[ sin(q)*sin(alpha), cos(q)*sin(alpha), cos(alpha), cos(alpha) * d],
[ 0, 0, 0, 1 ]
])
# Make Rotation Matrix
# x-axis
def rot_x(q):
return Matrix([[ 1, 0, 0],
[ 0, cos(q), -sin(q)],
[ 0, sin(q), cos(q)]])
# y-axis
def rot_y(q):
return Matrix([[ cos(q), 0, sin(q)],
[ 0, 1, 0],
[-sin(q), 0, cos(q)]])
# z-axis
def rot_z(q):
return Matrix([[ cos(q), -sin(q), 0],
[ sin(q), cos(q), 0],
[ 0, 0, 1]])
# Homegeneous Translatation Matrix
def to_homogenous(r, t = None):
"""
:r: Rotation Matrix
:t: Translation Matrix
:return: Homegeneous Matrix
"""
if t is None:
t = Matrix([0,0,0])
return r.row_join(t).col_join(Matrix([0,0,0,1]).T)
def test_code(test_case):
if len(test_case) <= 0: return
## Set up code
## Do not modify!
x = 0
class Position:
def __init__(self,EE_pos):
self.x = EE_pos[0]
self.y = EE_pos[1]
self.z = EE_pos[2]
class Orientation:
def __init__(self,EE_ori):
self.x = EE_ori[0]
self.y = EE_ori[1]
self.z = EE_ori[2]
self.w = EE_ori[3]
position = Position(test_case[0][0])
orientation = Orientation(test_case[0][1])
class Combine:
def __init__(self,position,orientation):
self.position = position
self.orientation = orientation
comb = Combine(position,orientation)
class Pose:
def __init__(self,comb):
self.poses = [comb]
req = Pose(comb) # position or orientation of the end effector
start_time = time()
########################################################################################
##
# Define variables
# α_{i-1} for Rx
alpha0, alpha1, alpha2, alpha3, alpha4, alpha5, alpha6 = symbols('alpha0:7')
# a_{i-1} for Dx
a0, a1, a2, a3, a4, a5, a6 = symbols('a0:7')
# θi for Rz, **Joint Angles**
q1, q2, q3, q4, q5, q6, q7 = symbols('q1:8')
# di for Dz
d1, d2, d3, d4, d5, d6, d7 = symbols('d1:8')
# Define dh parameters
dh_params = {
alpha0: 0, a0: 0, q1: q1, d1: 0.33+0.42,
alpha1: -pi/2, a1: 0.35, q2: q2-pi/2, d2: 0,
alpha2: 0, a2: 1.25, q3: q3, d3: 0,
alpha3: -pi/2, a3: -0.054, q4: q4, d4: 0.96+0.54,
alpha4: pi/2, a4: 0, q5: q5, d5: 0,
alpha5: -pi/2, a5: 0, q6: q6, d6: 0,
alpha6: 0, a6: 0, q7: 0, d7: 0.193+0.11,
}
# Make matrixies
t1 = matrix_from(alpha0, a0, q1, d1).subs(dh_params)
t2 = matrix_from(alpha1, a1, q2, d2).subs(dh_params)
t3 = matrix_from(alpha2, a2, q3, d3).subs(dh_params)
t4 = matrix_from(alpha3, a3, q4, d4).subs(dh_params)
t5 = matrix_from(alpha4, a4, q5, d5).subs(dh_params)
t6 = matrix_from(alpha5, a5, q6, d6).subs(dh_params)
t7 = matrix_from(alpha6, a6, q7, d7).subs(dh_params)
t_ee = t1 * t2 * t3 * t4 * t5 * t6 * t7
pos_ee = Matrix(test_case[0][0])
## Insert IK code here!
# the end effector -
# Quaternion to Euler angles(roll,pitch,yaw)
_ori = req.poses[0].orientation
angles = tf.transformations.euler_from_quaternion([_ori.x, _ori.y, _ori.z, _ori.w])
roll, pitch, yaw = symbols('roll pitch yaw')
# Generate rotation matrix
rot_fix = rot_z(pi) * rot_y(-pi/2) # Fix for URDF <-> DH-convertion
_rot_ee = rot_z(yaw) * rot_y(pitch) * rot_x(roll) * rot_fix
rot_ee = _rot_ee.subs({roll: angles[0], pitch: angles[1], yaw: angles[2]})
# Obtain the position of wrist center from the end effector's.
pos_wc = pos_ee - dh_params[d7] * rot_ee[:,2]
# Project the position to the plane, and thus obtain angle of (x, y)
theta1 = atan2(pos_wc[1], pos_wc[0])
# Obtain side lengths of fig3 using pythagorean theorem
fig3_side_a = pos_wc[0] # x
fig3_side_b = pos_wc[1] # y
fig3_side_c = sqrt(fig3_side_a ** 2 + fig3_side_b ** 2)
# Obtain side lengths of fig2 using pythagorean theorem
fig2_side_a = pos_wc[2] - dh_params[d1] # z - height of joint1
fig2_side_b = fig3_side_c - dh_params[a1] # side_b-offset
fig2_side_c = sqrt(fig2_side_a ** 2 + fig2_side_b ** 2)
# Obtain side lengths of fig1 using pythagorean theorem
fig1_side_a = dh_params[d4]
fig1_side_b = fig2_side_c
fig1_side_c = dh_params[a2]
## Obtain angles of fig1
#
# Get triangle angles using sides of triangle
#
# ref.low of cosines https://en.wikipedia.org/wiki/Law_of_cosines#Applications
def angles_from_triangle_sides(a, b, c):
""":a, b, c: side values of triangle"""
gamma = acos((a**2 + b**2 - c**2) / (2 * a * b))
beta = acos((a**2 + c**2 - b**2) / (2 * a * c))
alpha = acos((b**2 + c**2 - a**2) / (2 * b * c))
return (alpha, beta, gamma)
fig1_angle_a, fig1_angle_b, _ = angles_from_triangle_sides(
fig1_side_a, fig1_side_b, fig1_side_c)
theta2 = pi/2 - fig1_angle_a - atan2(fig2_side_a, fig2_side_b)
theta3 = pi/2 - fig1_angle_b + atan2(dh_params[a3], dh_params[d4])
# Slice Rotation Matrix, i=1..3
rot0_3 = (t1 * t2 * t3)[0:3, 0:3].evalf(
subs={q1: theta1, q2: theta2, q3: theta3})
# Obtain the Rotation Matrix, i=3..6
# Memo:
# r: rotation_matrix
# inv(r) == r^T
rot3_6 = rot0_3.T * rot_ee
# Obtain euler angles from rotation matrix
# ex. rot = t4 * t5 * t6
def euler_angles_from_rotation_matrix(rot):
# simplify(t4 * t5 * t6 * t7)[0:3,0:3]
# = [[*, *, -sin(q5)cos(q4)],
# [sin(q5)cos(q6), -sin(q5)sin(q6), cos(q5) ],
# [*, *, sin(q4)sin(q5) ]]
# atan2(sin(q4)sin(q5), cos(q4)sin(q5))
alpha = atan2(rot[2,2], -rot[0,2])
# atan2(sin(q5), cos(q5))
beta = atan2(sqrt(rot[0,2]**2 + rot[2,2]**2), rot[1,2])
# atan2(sin(q5)sin(q6), sin(q5)cos(q6))
gamma = atan2(-rot[1,1], rot[1,0])
return (alpha, beta, gamma)
theta4, theta5, theta6 = euler_angles_from_rotation_matrix(rot3_6)
##
########################################################################################
########################################################################################
## For additional debugging add your forward kinematics here. Use your previously calculated thetas
## as the input and output the position of your end effector as your_ee = [x,y,z]
## (OPTIONAL) YOUR CODE HERE!
fk = t_ee.evalf(subs={
q1: theta1, q2: theta2, q3: theta3,
q4: theta4, q5: theta5, q6: theta6,
})
## End your code input for forward kinematics here!
########################################################################################
## For error analysis please set the following variables of your WC location and EE location in the format of [x,y,z]
your_wc = pos_wc # <--- Load your calculated WC values in this array
your_ee = fk[:3,3] # <--- Load your calculated end effector value from your forward kinematics
########################################################################################
## Error analysis
print ("\nTotal run time to calculate joint angles from pose is %04.4f seconds" % (time()-start_time))
# Find WC error
if not(sum(your_wc)==3):
wc_x_e = abs(your_wc[0]-test_case[1][0])
wc_y_e = abs(your_wc[1]-test_case[1][1])
wc_z_e = abs(your_wc[2]-test_case[1][2])
wc_offset = sqrt(wc_x_e**2 + wc_y_e**2 + wc_z_e**2)
print ("\nWrist error for x position is: %04.8f" % wc_x_e)
print ("Wrist error for y position is: %04.8f" % wc_y_e)
print ("Wrist error for z position is: %04.8f" % wc_z_e)
print ("Overall wrist offset is: %04.8f units" % wc_offset)
# Find theta errors
t_1_e = abs(theta1-test_case[2][0])
t_2_e = abs(theta2-test_case[2][1])
t_3_e = abs(theta3-test_case[2][2])
t_4_e = abs(theta4-test_case[2][3])
t_5_e = abs(theta5-test_case[2][4])
t_6_e = abs(theta6-test_case[2][5])
print ("\nTheta 1 error is: %04.8f" % t_1_e)
print ("Theta 2 error is: %04.8f" % t_2_e)
print ("Theta 3 error is: %04.8f" % t_3_e)
print ("Theta 4 error is: %04.8f" % t_4_e)
print ("Theta 5 error is: %04.8f" % t_5_e)
print ("Theta 6 error is: %04.8f" % t_6_e)
print ("\n**These theta errors may not be a correct representation of your code, due to the fact \
\nthat the arm can have muliple positions. It is best to add your forward kinmeatics to \
\nconfirm whether your code is working or not**")
print (" ")
# Find FK EE error
if not(sum(your_ee)==3):
ee_x_e = abs(your_ee[0]-test_case[0][0][0])
ee_y_e = abs(your_ee[1]-test_case[0][0][1])
ee_z_e = abs(your_ee[2]-test_case[0][0][2])
ee_offset = sqrt(ee_x_e**2 + ee_y_e**2 + ee_z_e**2)
print ("\nEnd effector error for x position is: %04.8f" % ee_x_e)
print ("End effector error for y position is: %04.8f" % ee_y_e)
print ("End effector error for z position is: %04.8f" % ee_z_e)
print ("Overall end effector offset is: %04.8f units \n" % ee_offset)
if __name__ == "__main__":
# Change test case number for different scenarios
for i in test_cases:
print ('### case %d' % i)
test_code(test_cases[i])