MATLAB code¶

This MATLAB code is documented with Sphinx using the matlabdomain extension.

src¶

src module contains the following source code files:

src.main¶

Main function

Example on the application of Lie group integrators for the solution of the equations of motion of a multibody system, August 2022, Andrea Leone @ NTNU

Ref.: E. Celledoni, E. Çokaj, A. Leone, D. Murari, B. Owren. “Lie group integrators for mechanical systems”, International Journal of Computer Mathematics, 99:1, 58-88.

src/integrators¶

src.integrators.LieEuler(vector_field, exponential, action, p, h, sigma0, trajectory, t)¶

Lie-Euler time integrator (Runge-Kutta-Munthe-Kaas method of order 1)

Parameters:

vector_field – right hand side of the ODE [type: function handle]
exponential – exponential map from the Lie algebra to the Lie group [type: function handle]
action – Lie group action [type: function handle]
p – solution at time t_n [type: float, 3x14 matrix]
h – time step size [type: float]
sigma0 – initial value of the curve sigma on the Lie algebra [type: float, 30x1 vector]
trajectory – desired trajectory [type: function handle]
t – discrete time t_n [type: float]

Returns:

solution at time t_(n+1) [type: float, 3x14 matrix]

src.integrators.RKMK2Heun(vector_field, exponential, action, p, h, sigma0, trajectory, t)¶

Runge-Kutta-Munthe-Kaas time integrator order 2, based on the Heun scheme

Parameters:

vector_field – right hand side of the ODE [type: function handle]
exponential – exponential map from the Lie algebra to the Lie group [type: function handle]
action – Lie group action [type: function handle]
p – solution at time t_n [type: float, 3x14 matrix]
h – time step size [type: float]
sigma0 – initial value of the curve sigma on the Lie algebra [type: float, 30x1 vector]
trajectory – desired trajectory [type: function handle]
t – discrete time t_n [type: float]

Returns:

solution at time t_(n+1) [type: float, 3x14 matrix]

src.integrators.RKMK3(vector_field, exponential, action, p, h, sigma0, trajectory, t)¶

Runge-Kutta-Munthe-Kaas time integrator order 3

Parameters:

vector_field – right hand side of the ODE [type: function handle]
exponential – exponential map from the Lie algebra to the Lie group [type: function handle]
action – Lie group action [type: function handle]
p – solution at time t_n [type: float, 3x14 matrix]
h – time step size [type: float]
sigma0 – initial value of the curve sigma on the Lie algebra [type: float, 30x1 vector]
trajectory – desired trajectory [type: function handle]
t – discrete time t_n [type: float]

Returns:

solution at time t_(n+1) [type: float, 3x14 matrix]

src.integrators.RKMK4¶

Runge-Kutta-Munthe-Kaas time integrator order 4

Parameters:

vector_field – right hand side of the ODE [type: function handle]
exponential – exponential map from the Lie algebra to the Lie group [type: function handle]
action – Lie group action [type: function handle]
p – solution at time t_n [type: float, 3x14 matrix]
h – time step size [type: float]
sigma0 – initial value of the curve sigma on the Lie algebra [type: float, 30x1 vector]
trajectory – desired trajectory [type: function handle]
t – discrete time t_n [type: float]

Returns:

solution at time t_(n+1) [type: float, 3x14 matrix]

src.integrators.CFree4(f, action, exponential, h, p, trajectory, t)¶

Commutator-free time integrator of order 4

Parameters:

f – map f from the phase space (on which the vector field is defined) to the Lie algebra [type: function handle]
action – Lie group action [type: function handle]
exponential – exponential map from the Lie algebra to the Lie group [type: function handle]
h – time step size [type: float]
p – solution at time t_n [type: float, 3x14 matrix]
trajectory – desired trajectory [type: function handle]
t – discrete time t_n [type float]

Returns:

solution at time t_(n+1) [type: float, 3x14 matrix]

src/lie_group_functions¶

src.lie_group_functions.hat(v)¶

It returns the skew symmetric matrix A associated to the vector v, such that hat(a)b=axb for all 3-component vectors a and b, with “x” the cross product

Parameters:: v – vector with 3 components
Returns:: element of the Lie algebra so(3) (skew symmetric 3x3 matrix)

src.lie_group_functions.invhat(A)¶

Inverse of the hat function

Parameters:: A – element of the Lie algebra so(3) (skew symmetric 3x3 matrix)
Returns:: 3x1 column vector v

src.lie_group_functions.expSO3(x)¶

Exponential map on SO(3)

Parameters:: input – element of the lie algebra so(3), represented as a vector with 3 components.
Returns:: element of the group SO(3), i.e. 3x3 rotation matrix

src.lie_group_functions.expSE3(input)¶

Exponential map on SE(3)

Parameters:: input – element of the lie algebra se(3) represented as 6-component vector, i.e. as a pair (u,v) with with the 3-component vector u corresponding to a skew symmetric matrix hat(u) and the 3-component vector v corresponding to the translational part.
Returns:: element of the group SE(3), represented as a 3x4 matrix [A, b], with A 3x3 rotation matrix and b 3-component translation vector.

src.lie_group_functions.dexpinvSO3(v, input)¶

Inverse of the derivative of the exponential map on SO(3)

Parameters:

v – element in the lie algebra so(3), represented as a 3x1 vector
input – element in the lie algebra so(3), represented as a 3x1 vector

Returns:

inverse of dexp_v(input) as 3x1 vector

src.lie_group_functions.dexpinvSE3(sigma, input)¶

Inverse of the derivative of the exponential map on SE(3)

Parameters:

sigma – element in the lie algebra se(3), represented as a 6x1 vector
input – element in the lie algebra se(3), represented as a 6x1 vector

Returns:

inverse of dexp_sigma(input) as 6x1 vector

src/pre_post_processing¶

src.pre_post_processing.preprocess¶: Preprocessing : definition and inizialization of support variables to plot errors and trajectories

src.pre_post_processing.postprocess¶: Postprocessing: update of support variables to plot errors and real time plot of the tracking problem

src.pre_post_processing.plots¶: plots

src/control_functions¶

src.control_functions.controls(m1, m2, my, L1, L2, J1, J2, B, trajectory)¶: Main function used to define the control functions

src.control_functions.getUpar(m1, m2, my, L1, L2, B, trajectory)¶: Support function used to define the control functions and the vector field

src.control_functions.getUperp(m1, m2, my, L1, L2, B, trajectory)¶: Support function used to define the control functions and the vector field

src.control_functions.deriv1(v, vdot)¶

Support function used to define the control functions

Parameters:

v – vector
vdot – time derivative of vector v

Returns:

time derivative of the vector v divided by the norm of v

src.control_functions.deriv2(v, vdot, v2dot)¶

Support function used to define the control functions

Parameters:

v – vector
vdot – time derivative of vector v
v2dot – second time derivative of vector v

Returns:

time derivative of the output of deriv1

src.control_functions.deriv3(v, vdot, v2dot, v3dot)¶

Support function used to define the control functions

Parameters:

v – vector
vdot – time derivative of vector v
v2dot – second time derivative of vector v
v3dot – third time derivative of vector v

Returns:

time derivative of the output of deriv2

src.control_functions.deriv4(v, vdot, v2dot, v3dot, v4dot)¶

Support function used to define the control functions

Parameters:

v – vector
vdot – time derivative of vector v
v2dot – second time derivative of vector v
v3dot – third time derivative of vector v
v4dot – fourth time derivative of vector v

src.control_functions.derivhatq2w(q, w, qdot, wdot)¶: Support function used to define the control functions

src.control_functions.deriv2hatq2w(q, w, qdot, wdot, q2dot, w2dot)¶: Support function used to define the control functions

src.control_functions.deriv3hatq2w(q, w, qdot, wdot, q2dot, w2dot, q3dot, w3dot)¶: Support function used to define the control functions

src.control_functions.deriv4hatq2w(q, w, qdot, wdot, q2dot, w2dot, q3dot, w3dot, q4dot, w4dot)¶: Support function used to define the control functions