You will need to define an equation that defines the kinematic loop
constraint (holonomic constraint). This will look like 0 = f(coordinates,
lengths). If you can then solve this equation for the dependent coordinate,
then you can construct all of your position vectors in terms of only the
independent coordinates, using the solution to the constraint equation to
substitute in for the dep coordinate. If you can't analytically solve the
loop constraint (often the case), then you need to define all your position
vectors in terms of both the independent and dependent coordinates. You can
then take the time derivative of f to get a pseudo velocity constraint
which you can include as an extra kinematic differential equation.
Depending on what you want to do (forward sim, initial value problem,
linearization, etc) there are different things to do with this equation.
moorepants.info
+01 530-601-9791
On Thu, Nov 30, 2017 at 9:14 AM, Aaron Shatters <shatters.aaro...@gmail.com>
wrote:
> 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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