ESE 547 F’22
Problem Set 2
Computing and Visualizing Attractors and Basins via Lyapunov Functions
D. E. Koditschek February 13, 2022
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Released Tue., Jan. 18 Corrected, Wed., Feb. 2 Due Tue., Feb. 22 at 11:59pm
This problem set has three regular questions, each of which include some
components. These will add to the numerator of your score without adding to the denomina- tor. Recall from the course description document that the five scheduled problem sets count cumulatively toward 60% of your final grade. The breakdown of points for this specific prob- lem set are posted in the rubric details up at the Canvas portal where you will be submitting your work.
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1. Here, we reconsider the damped harmonic oscillator VF , fDHO(x) (2.1.7), characterized by the parameter values m = k = b = 1 as introduced in PS 1 Exercise 1. Since this problem set explores related properties in two different coordinate representations of the same (two-dimensional) state space, it will be convenient to name two different copies of the plane to distinguish physical (position and velocity) coordinates, x ∈ X := R2 from the abstract (real canonical) coordinates y ∈ Y := R2.
(a) Regenerate the ODE-solver from PS 1 Exercise 1-1a and compare the computed change in total energy along traejctories over time, ηHO (2.1.15), with the power function η ̇HO (2.1.16) as follows.
Plot all output graphs for solutions generated using the eight different initial con-
ditions x0 ∈ { cos θi : θi = iπ/4,i ∈ {0,…7}} ⊂ X. sin θi
Extra Credit
i. Present a “three dimensional plot” depicting the energy values, ηHO◦ft x0, DHO
as a curve of (nonnegative) “heights” over the statespace plane, X , on which you superimpose the forward time orbits, fR+ x0. Your output will look
something along the lines of Lecture Notes Figure 2.7
ii. Superimpose on that previous output an additional set of eight “three dimen-
sional curves” that plot the values of the power function, η ̇DHO ◦ft x0 as DHO
a curve of (nonpositive) “depths” below the statespace plane, X. Now your output will juxtapose a graphic along the lines of Lecture Notes Figure 3.5 ly- ing “below” the X -plane that appears to be supporting “above” it the previous graphic (Lecture Notes Figure 2.7).
iii. Compute a numerical estimate of the rate of energy change
η ̇HO (tk) := 1 ηHO ◦ ftk x0 − ηHO ◦ ftk−1 x0 ,
where k ranges over the numerical values of time that your simulation was allowed to run for each IC , x0, and ∆Tk denotes the time step the solver used to advance from the (k − 1)th to the kth time step of its computed solution.
Substitute η ̇HO for η ̇HO in the previous “three dimensional plot”.
iv. Plot the error between the closed form power function, η ̇HO, and its numeri-
cally computed estimate, η ̇HO,
ε(tk):=η ̇HO ◦ftk x0−η ̇HO(tk)
in a new “three dimensional plot” where each error value is plotted as a “height” over or under the statespace plane, X on which you superimpose the forward time orbits.
v. Extra Credit
Explain why ηHO cannot be used in conjunction with Lyapunov’s theorem (Lecture Notes Section 3.1.2.2) to mathematically prove that the origin of X is asymptotically stable under the flow generated by fDHO.
Instead, use ηHO in conjunction with Lasalle’s theorem (Lecture Notes Section 3.1.2.2) to prove mathematically that the origin is asymptotically stable.
(b) Although Lord Kelvin “knew” [1] that the PD property of ηHO coupled with the NSD property of η ̇HO was sufficient to conclude the asymptotic stability of the origin, it fell to Lasalle [2] to give a formal mathematical justification for this conclusion.
Here, we will use Lyapunov’s theorem by appeal to his generalized concept of energy [3] in the form of a different PD function that turns out to have a ND derivative.
Specifically, we will introduce a PD (generalized energy) function
ηRC :Y →R:y→ 21yTy=∥y∥2/2 (1.2)
whose associated power function, η ̇RC is ND along the motions of fRC (2.1.13), and use the CC hRC to “pull it back” into the x-coordinate plane where it can be compared to the physical total energy, ηHO.
i. Compute the closed form generalized power function, η ̇RC and simplify terms to show that it is a multiple of the squared Euclidean norm in the y-plane. Now use the code of the previous question, Exercise 1-1a, to redo those same steps (i. through iv.) to visualize ηRC & η ̇RC in plots over the y-coordinate plane, Y. Your plots should reveal that η ̇RC lies strictly “beneath” the y-plane, Y, except at the origin as required to be ND.
v. Now compute the conjugate of ηRC in physical coordinates x ∈ X = h−1 (Y)
Once again, repeat the same steps of Exercise 1-1a (i. through iv.) to visualize η ̃RC and η ̃ ̇RC in physical coordinates, x ∈ X .
ix. Explain how your plots corroborate the claim that ηRC is a strict LF for fDHO
x. Extra Credit
Verify mathematically that ηRC is really a strict LF for fDHO and use Lya- punov’s Theorem to prove that the origin of X is asymptotically stable.
2. This problem aims to provide a computational example of a dynamical (i.e. non- equilibrium) steady state behavior called a limit cycle that will become our archetype for understanding and commanding gaits.
We start with a nonlinear modification of fRC in the y-coordinate plane, Y,
fnRC(y) := [σ(∥y∥)I2 + ωJ2] y (1.3)
where ω is a constant as in (2.1.13) but σ is the scalar function σ(ρ) := 1 − ρ2.
and we will use reuse ηRC (1.2), but this time in a composition that accords it a role in the construction of a Lasalle function.
(a) Use the ODE solver of your choice to simulate the solutions of the dynamical system (1.3) generated by fnRC from the eight different initial conditions
y0 ∈{2γ(−1)βα,(−1)β(1−α)T :α,β∈{0,1} & γ∈{−1,1}}⊂Y
i. Plot the resulting trajectories as functions of time so that your outputs have the general appearance of the pair of left hand side traces in the panels of Figure 2.3.
η ̃ R C : X → R : x → η R C ◦ h R C ( x )
and use fDHO to compute its associated (generalized) power function
η ̃ ̇RC (x) = Dη ̃RC (x) · fDHO(x)
We don’t yet have good physical interpretation of this system, nevertheless, write a sentence or two that describes the empirical observation you would make about the state trajectories in contrast to the state trajectories of the flow generated by fRC.
ii. Plot the orbits of the flow through these eight initial conditions so that your output has the general appearance of Figure 2.4.
Describe qualitatively in a sentence or two the nature of the forward limit set of the entire state space plane, Y.
iii. Extra Credit
Plot the trajectories through these eight initial conditions as spatial curves of two-dimensional solution vectors parametrized by time. The output should look like a set of eight corkscrew curves in three-space that start on the state space plane at time t = 0 and wind around the time axis toward some desti- nation
(b) Given the generalized energy,
ηnRC(y) := 1 [σ ◦ 2ηRC(y)]2 = 1 σ yTy2 , (1.4)
compute the associated generalized power function along the motions generated by fnRC,
η ̇nRC (y) = DηnRC (y) · fnRC(y).
Once again, use the code of the earlier question, Exercise 1-1a, to redo those same steps (i. through iv.) to visualize ηnRC & η ̇nRC in the real canonical coordinate plane, y ∈ Y.
(c) Interpret the numerical evidence from your plots as suggesting that ηnRC is a
Lasalle function and use the associated theorem to conjecture a formula for the
forward limit set, f∞ Y. nRC
(d) Extra Credit
Prove mathematically that ηnRC is a Lasalle function and use the associated the-
orem to furnish a formula for the forward limit set f∞ Y. nRC
3. This problem aims to provide an advance computational familiarity with what we shall come to call the “active damping stance mode controller” for vertical hopping following a method presented in [4].
Specifically in Lecture Notes Section 5.3.2 (still in preparation for upload to the Canvas site) we will introduce a controlled version of the damped harmonic oscillator (5.3.23)
0ω0 fcDHO(x,τ)= −ω −2ζ ̄ω x+ τ/ω
where the parameter assignments, ω := 1 & ζ ̄ := 21 put the unforced system (i.e., where τ ≡ 0) into correspondence with the instance of fDHO we have been exploring numerically. Let the actuator input take the form (5.3.24) (again, to appear in Lecture Notes Section 5.3.2)
τ(x) := ktx2 ∥x∥+ε
where we will let kt := 1 (it can be an arbitrary positive gain) and ε := 1/100 (it can be any “sufficiently small” positive offset). We will denote the resulting closed loop VF
fAD(x) := fcDHO(x,τ(x)). Finally, define a Lasalle function candidate for fAD,
1 ̄ kt/2ω22 ηAD(x):=2 −ζ+∥x∥+ε
(a) Repeat the steps of Exercise 2-2a for fAD where the set of eight IC for this numerical study is given as
x0 ∈{ρ(−1)βα,(−1)β(1−α)T :α,β∈{0,1} & ρ∈{ρ−,ρ+}}⊂X
̄ kt/2ω2 ̄ kt/2ω2 ρ+ ∈{ρ∈R+ :ζ> ρ+ε }; ρ− ∈{ρ∈R+ :ζ< ρ+ε }.
(b) Given that these numerical simulations use physical coordinates, x ∈ X , describe in a sentence or two the intuitive steady state behavior of the mass when subject to the control τ(x) that yields the closed loop fAD.
(c) Repeat the steps of Exercise 2-2a for the unforced system, fcDHO(x, 0) with ζ ̄ := 0 (which coincides exactly with fDHO for b := 0) for the same initial conditions, x0 as just above, substituting ηHO, the total mechanical energy, in place of ηAD. Describe in a sentence or two the intuitive steady state behavior of the mass under these conditions, making sure to explain how it is different from that produced by fAD .
(d) Extra Credit
Show that ηAD is a Lasalle function for fAD and use Lasalle’s Theorem to give a
mathematical characterization of the forward limit set f∞ X. AD
2 Appendix - Tables of Notation
Abbreviated Terminology
VF Section 2.1.1
Description
vector field
initial condition
general change of coordinates linear change of coordinates fixed point of a VF
positive definite
Equation Description
(2.1.1) term := definition of term (2.1.2) definition of term =: term (2.1.9) matrix multiplication
Segment IC Section 2.1.1
Equation (2.1.3) (2.1.3) (2.1.11) (2.1.11)
CC Section 2.1.1 CCL Section 2.1.1 FP Section 2.2.2 PD Section 3.1.2
Generic Symbols and Spaces
hdiag fDHO ∼ fdiag Rn Rn×m
J2 e1, e2 g
Segment Section 2.1.1 Section 2.1.1 Section 2.1.2
Section 2.1.2 Section 2.1.2 Section 2.1.2 Section 2.1.2 Section 2.1.2 Section 2.1.2 Section 2.1.2 Section 2.1.3 Section 2.2.1 Section 2.2.1
(2.1.11) fdiag space space space space
conjugate to fDHO , via CC hdiag of n × 1 real vectors
of n × m real matrices
of n × 1 complex vectors
of n × m complex matrices (2.1.13) real component of a complex number (2.1.13) real component of a complex number
(2.1.15) function composition (2.2.1) Unit circle
(2.2.1) Unit circle tangent space
Segment Section 2.1.2 Section 2.1.2 Section 2.2.1 Section 2.2.1
Equation (2.1.13) (2.1.13)
Description
n × n identity matrix
2 × 2 skew symmetric matrix planar unit vectors gravitational constant
Variables & Functions
Segment Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1 Section 2.1.1
Equation (2.1.1)
(2.1.3) (2.1.4) (2.1.5) (2.1.6) (2.1.6) (2.1.7) (2.1.10) (2.1.11) (2.1.11) (2.1.11) (2.1.13) (2.1.15) (2.1.16)
Equation (2.2.1) (2.2.2) (2.2.2) (2.2.3) (2.2.4) (2.2.5)
Description
prismatic joint variable
scalar damper force function
scalar damper vector field
scalar damper flow
Hooke’s spring potential
prismatic tangent vector
damped harmonic oscillator acceleration damped harmonic oscillator vector field damped harmonic oscillator flow
diagonalizing linear coordinate transformation conjugate VF to f
diagonalized (complex) coordinates for x rational canonical (RC) coordinates for x harmonic oscillator total energy
damped harmonic oscillator power function
Description
revolute tangent vector
1 DoF Revolute Kinematics
Cartesian coordinates for pendulum point mass body Kinetic Energy for 1 DoF Revolute Kinematics Potential Energy for 1 DoF Revolute Kinematics Lagrangian for 1 DoF Revolute Kinematics
Tait. Treatise on natural philosophy. University
[2] J. P. LaSalle. The Stability of Dynamical Systems. Society for Industrial Mathematics,
[3] ̆ı Lyapunov. Problème général de la stabilité du mouvement. Princeton
University Press, 1949. 2
[4] A. De and D.E. Koditschek. Parallel composition of templates for tail-energized planar hopping. In 2015 IEEE International Conference on Robotics and Automation (ICRA), page 4562–4569. IEEE, May 2015. 4
Variables & Functions, ctd.
q Section 2.2.1
Segment gR Section 2.2.1
b Section 2.2.1 κR Section 2.2.1 φR Section 2.2.1 λR Section 2.2.1
References
[1] Sir William Thomson and of Cambridge, 1888. 2
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