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simple_ik_equations_numba.py
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simple_ik_equations_numba.py
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###########################################################################
# The following license applies to simple_ik.py and
# simple_ik_equations_numba.py (the "Files"), which contain software
# for use with the Stretch mobile manipulators, which are robots
# produced and sold by Hello Robot Inc.
# Copyright 2024 Hello Robot Inc.
# This software is free software: you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public License
# v3.0 (GNU LGPLv3) as published by the Free Software Foundation.
# This software is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License v3.0 (GNU LGPLv3) for more details,
# which can be found via the following link:
# https://www.gnu.org/licenses/lgpl-3.0.en.html
# For further information about the Files including inquiries about
# dual licensing, please contact Hello Robot Inc.
###########################################################################
from numba import njit
import math
import numpy as np
@njit
def idealized_fk_with_rotary_base(base_angle, lift_distance, arm_distance, b1):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
# lift (lambda)
L = lift_distance
# arm (alpha)
A = arm_distance
# base rotation (theta)
T = base_angle
x = (-math.cos(T) * b1) + (math.sin(T) * A)
y = (-math.sin(T) * b1) - (math.cos(T) * A)
z = L
return((x,y,z))
@njit
def calibrated_fk_with_rotary_base(base_angle, lift_distance, arm_distance, b1, l_vector, a_vector):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
# unit vector in the direction of positive lift motion
l1, l2, l3 = l_vector
# unit vector in the direction of positive arm extension
a1, a2, a3 = a_vector
# lift (lambda)
L = lift_distance
# arm (alpha)
A = arm_distance
# base rotation (theta)
T = base_angle
term1 = (A * a1) + (L * l1) + b1
term2 = (A * a2) + (L * l2)
x = (math.cos(T) * term1) - (math.sin(T) * term2)
y = (math.sin(T) * term1) + (math.cos(T) * term2)
z = (A * a3) + (L * l3)
return((x,y,z))
@njit
def idealized_ik_with_rotary_base(wrist_position, b1):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
x, y, z = wrist_position
# lift (lambda)
L = z
# arm (alpha)
A = math.sqrt(x**2 + y**2 - b1**2)
# base rotation (theta)
T = -2 * math.atan((a + y) / (b1 - x))
return((T, L, A))
@njit
def calibrated_ik_with_rotary_base(wrist_position, b1, l_vector, a_vector):
x, y, z = wrist_position
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
# unit vector in the direction of positive lift motion
l1, l2, l3 = l_vector
# unit vector in the direction of positive arm extension
a1, a2, a3 = a_vector
# arm (alpha)
A = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( -1 * l3 * ( a1 * x + a2 * y ) + a3 * ( l1 * x + l2 * y ) ) )**( -1 ) * ( ( a3 )**( 2 ) * ( l1 * x + l2 * y ) * ( b1 * l1 * l3 + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * z ) + ( -1 * a3 * l3 * ( a2 * b1 * l1 * l3 * y + ( a1 * b1 * l3 * ( 2 * l1 * x + l2 * y ) + ( a1 * ( 2 * ( l1 )**( 2 ) * x + ( ( l2 )**( 2 ) * x + l1 * l2 * y ) ) * z + a2 * ( l1 * l2 * x + ( ( l1 )**( 2 ) * y + 2 * ( l2 )**( 2 ) * y ) ) * z ) ) ) + l3 * ( ( a1 )**( 2 ) * b1 * ( l3 )**( 2 ) * x + ( a1 * a2 * b1 * ( l3 )**( 2 ) * y + ( ( a1 )**( 2 ) * l1 * l3 * x * z + ( a1 * a2 * l2 * l3 * x * z + ( a1 * a2 * l1 * l3 * y * z + ( ( a2 )**( 2 ) * l2 * l3 * y * z + ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) )
if A < 0.0:
# arm (alpha)
A = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( -1 * l3 * ( a1 * x + a2 * y ) + a3 * ( l1 * x + l2 * y ) ) )**( -1 ) * ( ( a3 )**( 2 ) * ( l1 * x + l2 * y ) * ( b1 * l1 * l3 + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * z ) + ( -1 * a3 * l3 * ( a2 * b1 * l1 * l3 * y + ( a1 * b1 * l3 * ( 2 * l1 * x + l2 * y ) + ( a1 * ( 2 * ( l1 )**( 2 ) * x + ( ( l2 )**( 2 ) * x + l1 * l2 * y ) ) * z + a2 * ( l1 * l2 * x + ( ( l1 )**( 2 ) * y + 2 * ( l2 )**( 2 ) * y ) ) * z ) ) ) + l3 * ( ( a1 )**( 2 ) * b1 * ( l3 )**( 2 ) * x + ( a1 * a2 * b1 * ( l3 )**( 2 ) * y + ( ( a1 )**( 2 ) * l1 * l3 * x * z + ( a1 * a2 * l2 * l3 * x * z + ( a1 * a2 * l1 * l3 * y * z + ( ( a2 )**( 2 ) * l2 * l3 * y * z + -1 * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) )
# lift (lambda)
L = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( -1 * l3 * ( a1 * x + a2 * y ) + a3 * ( l1 * x + l2 * y ) ) )**( -1 ) * ( -1 * ( a3 )**( 3 ) * b1 * l1 * ( l1 * x + l2 * y ) + ( -1 * ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) * ( a1 * x + a2 * y ) * z + ( ( a3 )**( 2 ) * ( a2 * b1 * l1 * l3 * y + ( a1 * b1 * l3 * ( 2 * l1 * x + l2 * y ) + ( -1 * a1 * l1 * ( l1 * x + l2 * y ) * z + -1 * a2 * l2 * ( l1 * x + l2 * y ) * z ) ) ) + a3 * ( -1 * ( a1 )**( 2 ) * b1 * ( l3 )**( 2 ) * x + ( -1 * a1 * a2 * b1 * ( l3 )**( 2 ) * y + ( 2 * ( a1 )**( 2 ) * l1 * l3 * x * z + ( ( a2 )**( 2 ) * l1 * l3 * x * z + ( a1 * a2 * l2 * l3 * x * z + ( a1 * a2 * l1 * l3 * y * z + ( ( a1 )**( 2 ) * l2 * l3 * y * z + ( 2 * ( a2 )**( 2 ) * l2 * l3 * y * z + ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) ) ) ) )
# base rotation (theta)
TArcTanArg1 = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( ( x )**( 2 ) + ( y )**( 2 ) ) )**( -1 ) * ( ( a3 )**( 2 ) * b1 * ( l2 )**( 2 ) * x + ( -2 * a2 * a3 * b1 * l2 * l3 * x + ( ( a2 )**( 2 ) * b1 * ( l3 )**( 2 ) * x + ( -1 * ( a3 )**( 2 ) * b1 * l1 * l2 * y + ( a2 * a3 * b1 * l1 * l3 * y + ( a1 * a3 * b1 * l2 * l3 * y + ( -1 * a1 * a2 * b1 * ( l3 )**( 2 ) * y + ( -1 * a2 * a3 * l1 * l2 * x * z + ( a1 * a3 * ( l2 )**( 2 ) * x * z + ( ( a2 )**( 2 ) * l1 * l3 * x * z + ( -1 * a1 * a2 * l2 * l3 * x * z + ( a2 * a3 * ( l1 )**( 2 ) * y * z + ( -1 * a1 * a3 * l1 * l2 * y * z + ( -1 * a1 * a2 * l1 * l3 * y * z + ( ( a1 )**( 2 ) * l2 * l3 * y * z + ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
TArcTanArg2 = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( ( x )**( 2 ) + ( y )**( 2 ) ) )**( -1 ) * ( ( -1 * l3 * ( a1 * x + a2 * y ) + a3 * ( l1 * x + l2 * y ) ) )**( -1 ) * ( ( a3 )**( 3 ) * b1 * l2 * ( ( l1 * x + l2 * y ) )**( 2 ) + ( -1 * ( a3 )**( 2 ) * ( l1 * x + l2 * y ) * ( a2 * b1 * l3 * ( l1 * x + 3 * l2 * y ) + ( a2 * l1 * ( l1 * x + l2 * y ) * z + -1 * a1 * l2 * ( -2 * b1 * l3 * x + ( l1 * x * z + l2 * y * z ) ) ) ) + ( -1 * l3 * ( ( a1 )**( 2 ) * a2 * b1 * ( l3 )**( 2 ) * ( x )**( 2 ) + ( 2 * a1 * ( a2 )**( 2 ) * b1 * ( l3 )**( 2 ) * x * y + ( ( a2 )**( 3 ) * b1 * ( l3 )**( 2 ) * ( y )**( 2 ) + ( ( a1 )**( 2 ) * a2 * l1 * l3 * ( x )**( 2 ) * z + ( -1 * ( a1 )**( 3 ) * l2 * l3 * ( x )**( 2 ) * z + ( 2 * a1 * ( a2 )**( 2 ) * l1 * l3 * x * y * z + ( -2 * ( a1 )**( 2 ) * a2 * l2 * l3 * x * y * z + ( ( a2 )**( 3 ) * l1 * l3 * ( y )**( 2 ) * z + ( -1 * a1 * ( a2 )**( 2 ) * l2 * l3 * ( y )**( 2 ) * z + ( -1 * a2 * x * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) + a1 * y * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) ) ) ) + a3 * ( ( a1 )**( 2 ) * l2 * l3 * x * ( b1 * l3 * x + -2 * ( l1 * x + l2 * y ) * z ) + ( ( a2 )**( 2 ) * l3 * y * ( b1 * l3 * ( 2 * l1 * x + 3 * l2 * y ) + 2 * l1 * ( l1 * x + l2 * y ) * z ) + ( 2 * a1 * a2 * l3 * ( b1 * l3 * x * ( l1 * x + 2 * l2 * y ) + ( ( l1 )**( 2 ) * ( x )**( 2 ) + -1 * ( l2 )**( 2 ) * ( y )**( 2 ) ) * z ) + ( -1 * l2 * x + l1 * y ) * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) )
else:
# lift (lambda)
L = -1 * ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( -1 * l3 * ( a1 * x + a2 * y ) + a3 * ( l1 * x + l2 * y ) ) )**( -1 ) * ( ( a3 )**( 3 ) * b1 * l1 * ( l1 * x + l2 * y ) + ( ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) * ( a1 * x + a2 * y ) * z + ( -1 * ( a3 )**( 2 ) * ( a2 * b1 * l1 * l3 * y + ( a1 * b1 * l3 * ( 2 * l1 * x + l2 * y ) + ( -1 * a1 * l1 * ( l1 * x + l2 * y ) * z + -1 * a2 * l2 * ( l1 * x + l2 * y ) * z ) ) ) + a3 * ( ( a1 )**( 2 ) * b1 * ( l3 )**( 2 ) * x + ( a1 * a2 * b1 * ( l3 )**( 2 ) * y + ( -2 * ( a1 )**( 2 ) * l1 * l3 * x * z + ( -1 * ( a2 )**( 2 ) * l1 * l3 * x * z + ( -1 * a1 * a2 * l2 * l3 * x * z + ( -1 * a1 * a2 * l1 * l3 * y * z + ( -1 * ( a1 )**( 2 ) * l2 * l3 * y * z + ( -2 * ( a2 )**( 2 ) * l2 * l3 * y * z + ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) ) ) ) )
# base rotation (theta)
TArcTanArg1 = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( ( x )**( 2 ) + ( y )**( 2 ) ) )**( -1 ) * ( ( a3 )**( 2 ) * b1 * ( l2 )**( 2 ) * x + ( -2 * a2 * a3 * b1 * l2 * l3 * x + ( ( a2 )**( 2 ) * b1 * ( l3 )**( 2 ) * x + ( -1 * ( a3 )**( 2 ) * b1 * l1 * l2 * y + ( a2 * a3 * b1 * l1 * l3 * y + ( a1 * a3 * b1 * l2 * l3 * y + ( -1 * a1 * a2 * b1 * ( l3 )**( 2 ) * y + ( -1 * a2 * a3 * l1 * l2 * x * z + ( a1 * a3 * ( l2 )**( 2 ) * x * z + ( ( a2 )**( 2 ) * l1 * l3 * x * z + ( -1 * a1 * a2 * l2 * l3 * x * z + ( a2 * a3 * ( l1 )**( 2 ) * y * z + ( -1 * a1 * a3 * l1 * l2 * y * z + ( -1 * a1 * a2 * l1 * l3 * y * z + ( ( a1 )**( 2 ) * l2 * l3 * y * z + -1 * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
TArcTanArg2 = ( ( ( a3 )**( 2 ) * ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) + ( -2 * a3 * ( a1 * l1 + a2 * l2 ) * l3 + ( ( a1 )**( 2 ) + ( a2 )**( 2 ) ) * ( l3 )**( 2 ) ) ) )**( -1 ) * ( ( ( x )**( 2 ) + ( y )**( 2 ) ) )**( -1 ) * ( ( -1 * l3 * ( a1 * x + a2 * y ) + a3 * ( l1 * x + l2 * y ) ) )**( -1 ) * ( ( a3 )**( 3 ) * b1 * l2 * ( ( l1 * x + l2 * y ) )**( 2 ) + ( -1 * ( a3 )**( 2 ) * ( l1 * x + l2 * y ) * ( a2 * b1 * l3 * ( l1 * x + 3 * l2 * y ) + ( a2 * l1 * ( l1 * x + l2 * y ) * z + -1 * a1 * l2 * ( -2 * b1 * l3 * x + ( l1 * x * z + l2 * y * z ) ) ) ) + ( -1 * l3 * ( ( a1 )**( 2 ) * a2 * b1 * ( l3 )**( 2 ) * ( x )**( 2 ) + ( 2 * a1 * ( a2 )**( 2 ) * b1 * ( l3 )**( 2 ) * x * y + ( ( a2 )**( 3 ) * b1 * ( l3 )**( 2 ) * ( y )**( 2 ) + ( ( a1 )**( 2 ) * a2 * l1 * l3 * ( x )**( 2 ) * z + ( -1 * ( a1 )**( 3 ) * l2 * l3 * ( x )**( 2 ) * z + ( 2 * a1 * ( a2 )**( 2 ) * l1 * l3 * x * y * z + ( -2 * ( a1 )**( 2 ) * a2 * l2 * l3 * x * y * z + ( ( a2 )**( 3 ) * l1 * l3 * ( y )**( 2 ) * z + ( -1 * a1 * ( a2 )**( 2 ) * l2 * l3 * ( y )**( 2 ) * z + ( a2 * x * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) + -1 * a1 * y * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) ) ) ) ) ) + a3 * ( ( a1 )**( 2 ) * l2 * l3 * x * ( b1 * l3 * x + -2 * ( l1 * x + l2 * y ) * z ) + ( ( a2 )**( 2 ) * l3 * y * ( b1 * l3 * ( 2 * l1 * x + 3 * l2 * y ) + 2 * l1 * ( l1 * x + l2 * y ) * z ) + ( 2 * a1 * a2 * l3 * ( b1 * l3 * x * ( l1 * x + 2 * l2 * y ) + ( ( l1 )**( 2 ) * ( x )**( 2 ) + -1 * ( l2 )**( 2 ) * ( y )**( 2 ) ) * z ) + ( l2 * x + -1 * l1 * y ) * ( ( ( l3 * ( a1 * x + a2 * y ) + -1 * a3 * ( l1 * x + l2 * y ) ) )**( 2 ) * ( ( a3 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l2 )**( 2 ) + ( ( l1 )**( 2 ) + ( l2 )**( 2 ) ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) ) + ( 2 * a1 * a2 * l2 * z * ( b1 * l3 + l1 * z ) + ( ( a2 )**( 2 ) * ( -1 * ( b1 )**( 2 ) * ( l3 )**( 2 ) + ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + ( -2 * b1 * l1 * l3 * z + -1 * ( l1 )**( 2 ) * ( z )**( 2 ) ) ) ) + ( ( a1 )**( 2 ) * ( ( l3 )**( 2 ) * ( ( x )**( 2 ) + ( y )**( 2 ) ) + -1 * ( l2 )**( 2 ) * ( z )**( 2 ) ) + 2 * a3 * ( a2 * l2 * ( ( b1 )**( 2 ) * l3 + ( -1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * l1 * z ) ) + -1 * a1 * ( l1 * l3 * ( ( x )**( 2 ) + ( y )**( 2 ) ) + b1 * ( l2 )**( 2 ) * z ) ) ) ) ) ) )**( 1/2 ) ) ) ) ) ) )
T = np.arctan2(TArcTanArg2, TArcTanArg1)
return((T, L, A))
@njit
def idealized_fk_with_prismatic_base(base_distance, lift_distance, arm_distance, b1):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
# lift (lambda)
L = lift_distance
# arm (alpha)
A = arm_distance
# base translation (mu)
M = base_distance
x = M - b1
y = -A
z = L
return((x,y,z))
@njit
def calibrated_fk_with_prismatic_base(base_distance, lift_distance, arm_distance, b1, l_vector, a_vector):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
# unit vector in the direction of positive lift motion
l1, l2, l3 = l_vector
# unit vector in the direction of positive arm extension
a1, a2, a3 = a_vector
# lift (lambda)
L = lift_distance
# arm (alpha)
A = arm_distance
# base translation (mu)
M = base_distance
x = A * a1 + L * l1 + M + b1
y = A * a2 + L * l2
z = A * a3 + L * l3
return((x,y,z))
@njit
def idealized_ik_with_prismatic_base(wrist_position, b1):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
x, y, z = wrist_position
# base translation (mu)
M = x + b1
# lift (lambda)
L = z
# arm (alpha)
A = -y
return((M, L, A))
@njit
def calibrated_ik_with_prismatic_base(wrist_position, b1, l_vector, a_vector):
# b1 is the distance in the x direction between the mobile base rotational axis to the lift on the ground
x, y, z = wrist_position
# unit vector in the direction of positive lift motion
l1, l2, l3 = l_vector
# unit vector in the direction of positive arm extension
a1, a2, a3 = a_vector
denominator = (a2 * l3) - (a3 * l2)
# base translation (mu)
M = (a1 * ((l2 * z) - (l3 * y)) +
a2 * ((-b1 * l3) - (l1 * z) + (l3 * x)) +
a3 * ((b1 * l2) + (l1 * y) - (l2 * x))) / denominator
# lift (lambda)
L = ((a2 * z) - (a3 * y)) / denominator
# arm (alpha)
A = ((-l2 * z) + (l3 * y)) / denominator
return((M, L, A))