pycapacity.algorithms module

Overview

There are three main polytope evaluation algorithms implemented:

  • hyper_plane_shift_method: Hyper plane shifting method implementation used to find half-space representation of problems of a form

\[P = \{x~ |~ y = Ax,\quad x_{min} <= x <= x_{max}\}\]
\[P = \{x~ |~ Ax = b,\quad b_{min} \leq b \leq b_{max}\}\]
\[P = \{x~ |~ Ax = By,\quad y_{min} \leq y \leq y_{max}\}\]

Additionally this module implements different helping functions to half-plane and vertex representation manipulations of the polytopes:

  • hspace_to_vertex: Function transforming H-representation to V-representation

  • vertex_to_hspace: Function transforming V-representation to H-representation

  • vertex_to_faces: Helping function transforming vertices of the polytope to a triangulated faces

pycapacity.algorithms.chebyshev_ball(A, b)[source]

Calculating chebyshev ball of a polytope given the half-space representation

https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.HalfspaceIntersection.html#r9b902253b317-1

Parameters:

  • A (list) – matrix of half-space representation Ax<b

  • b (list) – vector of half-space representation Ax<b

Returns:

returns a chebyshev center of the polytope radius(float): returns a chebyshev radius of the polytope

Return type:

center(array)

pycapacity.algorithms.chebyshev_center(A, b)[source]

Calculating chebyshev center of a polytope given the half-space representation

https://pageperso.lis-lab.fr/~francois.denis/IAAM1/scipy-html-1.0.0/generated/scipy.spatial.HalfspaceIntersection.html

Parameters:

  • A (list) – matrix of half-space representation Ax<b

  • b (list) – vector of half-space representation Ax<b

Returns:

returns a chebyshev center of the polytope

Return type:

center(array)

pycapacity.algorithms.face_index_to_vertex(vertices, indexes)[source]

Helping function for transforming the list of faces with indexes to the vertices

Parameters:

  • vertices – list of vertices

  • indexes – list of vertex indexes forming faces

Returns:

list of faces composed of vertices

Return type:

faces

pycapacity.algorithms.hspace_to_vertex(H, d)[source]

From half-space representation to the vertex representation

Parameters:

  • H (list) – matrix of half-space representation Hx<d

  • d (list) – vector of half-space representation Hx<d

Returns:

  • vertices(list) (vertices of the polytope)

  • face_indexes(list) (indexes of vertices forming triangulated faces of the polytope)

pycapacity.algorithms.hyper_plane_shift_method(A, x_min, x_max, tol=1e-15)[source]

Hyper plane shifting method implementation used to solve problems of a form:

\[P = \{y~ |~y = Ax, \quad x_{min} \leq x \leq x_{max}\}\]

Note

Gouttefarde M., Krut S. (2010) Characterization of Parallel Manipulator Available Wrench Set Facets. In: Lenarcic J., Stanisic M. (eds) Advances in Robot Kinematics: Motion in Man and Machine. Springer, Dordrecht

This algorithm can be used to calculate acceleration polytope, velocity polytoe and even polytope of the joint achievable joint torques based on the muscle forces

Parameters:

  • A – projection matrix

  • x_min – minimal values

  • x_max – maximal values

Returns:

  • H(list) – matrix of half-space representation Hx<d

  • d(list) – vector of half-space representation Hx<d

Raises:

  • ValueError – if the limits are not of the same size as the projection matrix

  • ValueError – if the limits are not of the same size

pycapacity.algorithms.iterative_convex_hull_method(A, B, y_min, y_max, tol, P=None, bias=None, G_in=None, h_in=None, G_eq=None, h_eq=None, max_iter=1000, verbose=False)[source]

A function calculating the polytopes of achievable x for equations form:

\[P_x = \{x~ |~ Ax = By,\quad y_{min} \leq y \leq y_{max}\}\]

(optionally - additional projection matrix)

\[P_x = \{x~ |~Pz = x,~~ Az = By,\quad y_{min} \leq y \leq y_{max}\}\]

(optionally - additional bias \(b\))

\[P_x = \{x~ |~Ax = By + b, \quad y_{min} \leq y \leq y_{max}\}\]

(optionally - additional inequality constaints)

\[G_{in} y \leq h_{in}\]

(optionally - additional equality constaints)

\[G_{eq} y = h_{eq}\]

Finally, the most general equation supported is

\[P_x = \{x~ |~Pz = x,~~ Az = By,~~ G_{in} y \leq h_{in} ,~~ G_{eq} y = h_{eq}, ~~ y_{min} \leq y \leq y_{max}\}\]

Note

On-line feasible wrench polytope evaluation based on human musculoskeletal models: an iterative convex hull method A.Skuric,V.Padois,N.Rezzoug,D.Daney

Parameters:

  • A – matrix

  • B – matrix

  • y_min – minimal values

  • y_max – maximal values

  • tol – tolerance for the polytope calculation

  • P – an additional projection matrix

  • bias – bias in the intermediate space

  • G_in – matrix - inegality constraint G_in y < h_in

  • h_in – vector - inegality constraint G_in y < h_in

  • G_eq – matrix - equality constraint G_eq y = h_eq

  • h_eq – vector - equality constraint G_eq y = h_eq

  • max_iter – maximum number of iterations (number of linera programmes solved) - default 1000

  • verbose – verbose program output - showing execution warnings - default False

Returns:

  • x_vert(list) – list of vertices

  • H(list) – matrix of half-space representation Hx<d

  • d(list) – vector of half-space representation Hx<d

  • faces(list) – list of vertex indexes forming polytope faces

  • y_vert(list) – list of y values producing x_vert

  • z_vert(list) – list of z values producing x_vert

Raises:

  • ValueError – if the dimensions of the input matrices are not valid

  • ValueError – if the limits dimensions are not valid

  • ValueError – if the limits min and max are not valid

  • ValueError – if the inequality/equality constraints dimensions are not valid

  • ValueError – if the the projection matrix dimensions are not valid

pycapacity.algorithms.order_index(points)[source]

Order clockwise 2D points

Parameters:

points – matrix of 2D points

Returns

indexes(array) : ordered indexes

pycapacity.algorithms.rotation_matrix_from_vectors(vec1, vec2)[source]

Find the rotation matrix that aligns vec1 to vec2

See https://stackoverflow.com/questions/45142959/calculate-rotation-matrix-to-align-two-vectors-in-3d-space

Parameters:

  • vec1 – A 3d β€œsource” vector

  • vec2 – A 3d β€œdestination” vector

Return mat:

A transform matrix (3x3) which when applied to vec1, aligns it with vec2.

pycapacity.algorithms.stack(A, B, dir='v')[source]

Helping function enabling vertical and horizontal stacking of numpy arrays

pycapacity.algorithms.vertex_enumeration_vepoli2(A, b_max, b_min, b_bias=None)[source]

Efficient vertex enumeration algorithm for a problem of a form:

\[P = \{x~ |~Ax = b, \quad b_{min} \leq b \leq b_{max}\}\]

Optional (bias \(b_{bias}\) can be added):

\[P = \{x~ |~Ax = b - b_{bias}, \quad b_{min} \leq b \leq b_{max}\}\]

Note

On-line force capability evaluation based on efficient polytope vertex search by A. Skuric, V. Padois, D. Daney

Parameters:

  • A – system matrix A

  • b_max – maximal b

  • b_min – minimal b

  • b_bias – b bias vector ( offset from 0 )

Returns:

  • x_vertex(list) (vertices of the polytope)

  • b_vartex(list) (b values producing x_vertex)

Raises:

  • ValueError – if the A matrix dimensions are not valid

  • ValueError – if the b_min and b_max are not of the same size as the A matrix

  • ValueError – if the b_bias is not of the same size as the A matrix

pycapacity.algorithms.vertex_to_faces(vertex)[source]

Function grouping the vertices to faces using a ConvexHull algorithm

Parameters:

vertex (array) – list of vertices

Returns:

list of triangle faces with vertex indexes which form them

Return type:

faces(array)

pycapacity.algorithms.vertex_to_hspace(vertex)[source]

Function transforming vertices to half-space representation using a ConvexHull algorithm

Parameters:

vertex (array) – list of vertices

Returns:

  • H(list) (matrix of half-space representation Hx<d)

  • d(list) (vector of half-space representation Hx<d)