I have a simple 3 member actuator where one of the members is a linear
actuator. I have a reference frame for each member (ground, arm, linear
actuator). I cannot seem to get sympy to use the point locations within
multiple frames to serve as constraints that will determine the relative
frame orientations. For example, if I have a triangle where each side of
the triangle has its own reference frame, the point positions alone should
be able to determine the frame orientations, but I cannot figure out how to
let sympy know this. How would I do this? (this generic pattern should
apply to any complex linkage with any number of closed geometries.)
I have included code:
from sympy import *
from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, Point
#
# I would like to develop the model for the below picture...
#
# There is an 'arm' and a linear 'actuator'.
# - the a_pt of the arm is connected to the ground
# - the y_pt of the actuator is connected to the ground
# - the k_pt is on the arm and the actuator is connected to it
#
# The desire is to solve the relationship between the length of the
actuator (a dynamic symbol)
# and the angle from vector AY to AB, as well as the magnitude of the Y-B
vector
#
# Since a closed path triangle is created between A, K, and Y, we should be
able to 'automagically'
# orient the frames with respect to each other, but I don't know how to do
this without introducing
# a bunch of dummy dynamic symbols to represent the orientations between
the frames.
#
# O <-- b_pt
# //
# //
# ---> //
# / //
# Arm //
# / O <-- k_pt
# ---> // \
# // \
# // \ <-- Actuator (extends/retracts)
# // \
# // \
# a_pt --> O============O <-- y_pt
# / / / /
# Ground
#
# Symbols for the Ground Points
ground_coord_a = symbols(('ground_coord_a_x', 'ground_coord_a_y'))
ground_coord_y = symbols(('ground_coord_y_x', 'ground_coord_y_y'))
# Symbols for the Arm Points
arm_coord_a = symbols(('arm_coord_a_x', 'arm_coord_a_y'))
arm_coord_k = symbols(('arm_coord_k_x', 'arm_coord_k_y'))
arm_coord_b = symbols(('arm_coord_b_x', 'arm_coord_b_y'))
# Symbol for the Actuator Cylinder Length
actuator_len = dynamicsymbols('actuator_len')
# Create the reference frame for each member
ground_frame = ReferenceFrame('ground_frame')
ground_frame_origin_pt = Point('ground_frame_origin_pt')
arm_frame = ReferenceFrame('arm_frame')
arm_frame_origin_pt = Point('arm_frame_origin_pt')
actuator_frame = ReferenceFrame('actuator_frame')
# Define all of the Points on the Linkage
a_pt = Point('a_pt')
b_pt = Point('b_pt')
k_pt = Point('k_pt')
y_pt = Point('y_pt')
# Place the points on the Ground
a_pt.set_pos(ground_frame_origin_pt, ground_coord_a[0] * ground_frame.x +
ground_coord_a[1] * ground_frame.y)
y_pt.set_pos(ground_frame_origin_pt, ground_coord_y[0] * ground_frame.x +
ground_coord_y[1] * ground_frame.y)
# Place the points on the Arm
a_pt.set_pos(arm_frame_origin_pt, arm_coord_a[0] * arm_frame.x + arm_coord_a
[1] * arm_frame.y)
b_pt.set_pos(arm_frame_origin_pt, arm_coord_b[0] * arm_frame.x + arm_coord_b
[1] * arm_frame.y)
k_pt.set_pos(arm_frame_origin_pt, arm_coord_k[0] * arm_frame.x + arm_coord_k
[1] * arm_frame.y)
# Place the points on the Actuator
k_pt.set_pos(y_pt, actuator_len * actuator_frame.x + 0.0 * actuator_frame.y)
# Of course things work within the same frame
length_ab = b_pt.pos_from(a_pt).magnitude()
print(length_ab)
# How about length Y to K?
length_yk = k_pt.pos_from(y_pt).magnitude()
print(length_yk)
# But not Y to B?
try:
length_yb = b_pt.pos_from(y_pt).magnitude()
print(length_yb)
except ValueError as e:
print(e)
# Ok, I can add a redundant symbol to orient the ground frame to the arm
frame...
# Dynamic Symbols (time variant)
angle_ground_to_arm = dynamicsymbols('angle_ground_to_arm')
arm_frame.orient(ground_frame, 'Axis', (angle_ground_to_arm, ground_frame.z
))
# Now can I get Y to B? still no
try:
length_yb = b_pt.pos_from(y_pt).magnitude()
print(length_yb)
except ValueError as e:
print(e)
# Ok, do I need to orient the actuator to the ground as well?
angle_ground_to_actuator = dynamicsymbols('angle_ground_to_actuator')
actuator_frame.orient(ground_frame, 'Axis', (angle_ground_to_actuator,
ground_frame.z))
# Now can I get Y to B?
# Yes, but now I have an expression that involves
'angle_ground_to_actuator(t)' as well as 'actuator_len(t)'
# which is redundant. I don't actually have to degrees of freedom or
actuators.
try:
length_yb = b_pt.pos_from(y_pt).magnitude()
print(length_yb)
except ValueError as e:
print(e)
# How do I make sympy know about the closed path so that it can 'infer'
# the frame orientations based on that closed path? (Without adding
redundant dummy dynamic symbols)
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