Dynamics and Control of a 3D
Pendulum
N. Harris McClamroch
Department of Aerospace Engineering
University of Michigan
4th February, 2005
Acknowledgements:
Dennis Bernstein
Sangbum Cho, Jinglai Shen, Amit Sanyal, Nalin A. Chaturvedi,
Fabio Bacconi, Mario Santillo, Taeyoung Lee
National Science Foundation
Outline of the Presentation
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Background: planar and spherical pendula, Lagrange top
What is a 3D pendulum?
3D pendulum models
Axially symmetric 3D pendulum
l Free dynamics of 3D axially symmetric pendulum
l Asymptotic stabilization of 3D axially symmetric pendulum
Asymmetric 3D Pendulum
l Free dynamics of 3D asymmetric pendulum
l Asymptotic stabilization of 3D asymmetric pendulum
Computational issues for 3D pendulum
Some experimental results for the 3D pendulum
Concluding remarks
Planar Pendulum
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One degree of freedom
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Global configuration space is S1
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Global phase space is S1 x R
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Total energy is conserved
Spherical Pendulum
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Two degrees of freedom
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Global configuration space is S2
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Global phase space is S2 ¥ R2
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Total energy is conserved
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Angular momentum component
about the vertical is conserved
Lagrange Top
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2 degrees of freedom
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Global configuration space is S2
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Global phase space is S2 ¥ R2
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Total energy is conserved
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Angular momentum component about
the vertical is conserved
What is a 3D Pendulum?
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Rigid body supported by a fixed,
frictionless pivot
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Three degrees of rotational
freedom
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Gravitational forces; disturbance
and control forces or moments
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Uniform constant gravity
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Generalization of the planar
pendulum, the spherical pendulum
and the Lagrange top
Motivation for Study of
3D Pendulum
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3D pendulum models are proposed as benchmarks for studying
nonlinear dynamics
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3D pendulum models are proposed as benchmarks for studying
nonlinear control (nonlinear stabilization)
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Pendulum models are representative of attitude systems in many
engineering applications, e.g. spacecraft in gravitational field and
robot systems
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Many interesting control problems for pendulum models in the
published literature
l Astrom and Furuta, Automatica,1996
l Egeland, et al, 1999, 2004
l Furuta, CDC 2003
Motivation
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3D pendulum may exhibit coupling between all three degrees of
freedom
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Does the 3D pendulum exhibit any new nonlinear dynamics that
have not been previously investigated?
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Does the 3D pendulum present any new nonlinear control design
issues that have not been previously investigated?
Triaxial Attitude Control Testbed
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University of Michigan
Experimental Facility: Triaxial
Attitude Control Testbed (TACT)
Experimental testbed for attitude
dynamics and control problems
TACT papers (McClamroch,
Bernstein)
TACT is a physical
implementation of the 3D
pendulum
Used for experimental validation
of theoretical results
Equations for 3D Pendulum
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Euler-Poisson equations
Evolution on SO(3) x R3
Global attitude representation
Reduced Equations for 3D Pendulum
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Define G = RT e3 ; then
Jw˙ = Jw ¥ w + mgr ¥ G
G˙ = G ¥ w
†
l G is direction of gravity in body frame
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G always has unity norm
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Reduced Euler-Poisson equations
Evolution on S2 x R3
Global reduced attitude representation
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Axially Symmetric 3D Pendulum
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Assumptions
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3D pendulum has a single axis of mass symmetry
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Body fixed axes are principal axes so that
J = diag(Jt, Jt, Ja )
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The pivot point and the center of mass lie on the axis of
symmetry so that
r = (0, 0, rs) where rs > 0
Uncontrolled Axially Symmetric
3D Pendulum
Thus
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Dynamics of axially symmetric 3D pendulum evolve
on S2 x R2
Uncontrolled Axially Symmetric
3D Pendulum
†
1 T
w Jw - mgr T G is conserved
2
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Total energy
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Angular momentum component about the vertical axis
w T JG †is conserved
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Terminology
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If c=0, the 3D pendulum is identical to the spherical pendulum
although the described global model is different from prior
models in the published literature
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If c≠0, the 3D pendulum is identical to the Lagrange top
Equilibria of Uncontrolled Axially
Symmetric 3D Pendulum
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The reduced equilibria of the axially symmetric 3D pendulum are
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If c = 0, the spherical pendulum has two distinct equilibria
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If c ≠ 0, the Lagrange top has two distinct relative equilibria
Stability of Uncontrolled Dynamics of
Axially Symmetric 3D Pendulum
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Local stability of the axially symmetric 3D pendulum on S2 x R2
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Well known stability results are as follows:
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The hanging reduced equilibrium of the spherical pendulum is always
locally stable
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The inverted reduced equilibrium of the spherical pendulum is always
locally unstable
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The hanging reduced equilibrium of the Lagrange top is always locally
stable
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The inverted reduced equilibrium of the Lagrange top is stable if and
only if
2 mgr s J t
c ≥
Ja
†
Simulation of Uncontrolled Dynamics
of Lagrange Top
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Small perturbations
from hanging
equilibrium
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Nonlinear oscillations
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Chaotic solutions are
not possible
Simulation of Uncontrolled Dynamics
of Lagrange Top
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Small
perturbations
from the inverted
equilibrium
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Nonlinear
oscillations
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Chaotic solutions
are not possible
Asymptotic Stabilization of Axially
Symmetric 3D Pendulum
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Controlled axially symmetric 3D pendulum
J t w˙ x = c(J t - J a )w y - mgr sGy + u x
J t w˙ y = c(J a - J t )w x + mgr sGx + u y
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†
G˙ x = Gy c - Gzw y
G˙ y = Gzw x - Gx c
G˙ z = Gx w y - Gy w x
Use feedback to asymptotically stabilize a reduced equilibrium of the
axially symmetric 3D pendulum
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Angular velocity feedback
u : R2 Æ R2
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Angular velocity and reduced attitude feedback
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u : S 2 ¥ R2 Æ R2
Existence of Smooth Stabilizing
Controllers for Axially Symmetric
3D Pendulum
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Global asymptotic stabilization of a reduced attitude
equilibrium of the axially symmetric 3D pendulum is not
possible using any continuous feedback
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Reason: there is a topological impediment due to the
fact that the configuration space S2 is compact
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What are possible closed loop domains of attraction?
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The domain of attraction may be all of S2 x R2 except
for a set of Lebesgue measure zero; this is referred to
as almost global asymptotic stabilization
Asymptotic Stabilization of Axially
Symmetric 3D Pendulum
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Asymptotic stabilization of the Lagrange top
l The hanging reduced equilibrium
l The inverted reduced equilibrium
l Some other reduced equilibrium
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Asymptotic stabilization of the spherical pendulum
l The hanging reduced equilibrium
l The inverted reduced equilibrium
l Some other reduced equilibrium
Asymptotic Stabilization of Hanging
Equilibrium of Lagrange Top:
Angular Velocity Feedback
Assume c ≠ 0. Consider the controller with
feedback of angular velocity
u x = -k x w x
u y = -k y w y
where kx > 0, ky > 0 .
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Theorem: The hanging equilibrium (Gh,0) is
asymptotically stable equilibrium of the closed loop
with a dense domain of attraction (S2 x R2 ) \ M.
Asymptotic Stabilization of Hanging
Equilibrium of Lagrange Top
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Controller arises from control Liapunov
function V : S 2 ¥ R 2 Æ R
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Controller is globally defined
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Angular velocity feedback only
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Obtain a nearly global domain of attraction of
the hanging equilibrium; must exclude the
stable manifold of the inverted equilibrium
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Simulation Results
Simulation Results
Simulation Results
Asymptotic Stabilization of Hanging
Equilibrium of Lagrange Top: Angular
Velocity and Reduced Attitude Feedback
Assume c ≠ 0. Consider the controller with feedback
of angular velocity and reduced attitude
u x = -(1+ k x J t Gzs h (G))w x - c(J t - J a )w y
Ê
Ê s h2 (G) ˆˆ
+ k x (cJ t Gx - Gy )s h (G) + Á mgr s - s h (G)f ' Á
˜˜Gy
4
Ë
¯¯
Ë
u y = -(1+ k y J t Gzs h (G))w y - c(J a - J t )w x
Ê
Ê s h2 (G) ˆˆ
+ k y (cJ t Gy - Gx )s h (G) - Á mgr s - s h (G)f ' Á
˜˜Gx
4
Ë
¯¯
Ë
where
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†
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s h (G) = 1- GhT G , kx > 0, ky > 0
f :[0,1) Æ R,strictly monotone
f (0) = 0, f (x) Æ • as x Æ1
Asymptotic Stabilization of Hanging
Equilibrium of Lagrange Top: Angular
Velocity and Reduced Attitude Feedback
Theorem: The hanging equilibrium (Gh,0) is
asymptotically stable equilibrium of the closed
loop with a dense domain of attraction (S2 \
{Gi} x R2).
Asymptotic Stabilization of Hanging
Equilibrium of Lagrange Top
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Controller arises from control Liapunov function V : S 2 ¥ R 2 Æ R
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Controller is not globally defined: not defined at
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inverted configuration
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The controller is PD feedback: angular velocity
feedback and reduced attitude feedback
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Obtain a nearly global domain of attraction of the
hanging equilibrium; must exclude the inverted
configuration
Simulation Results
Simulation Results
Simulation Results
Asymptotic Stabilization of Inverted
Equilibrium of Lagrange Top
Assume c ≠ 0. Consider the controller with feedback
of angular velocity and reduced attitude
u x = -(1+ k x J t Gzs i (G))w x - c(J t - J a )w y
Ê
Ê s i2 (G) ˆˆ
- k x (cJ t Gx - Gy )s i (G) + Á mgr s + s i (G)f ' Á
˜˜Gy
4
Ë
¯¯
Ë
u y = -(1+ k y J t Gzs i (G))w y - c(J a - J t )w x
Ê
Ê s i2 (G) ˆˆ
- k y (cJ t Gy - Gx )s i (G) - Á mgr s + s i (G)f ' Á
˜˜Gx
4
Ë
¯¯
Ë
where
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s i (G) = 1- GiT G , kx > 0, ky > 0
f :[0,1) Æ R,strictly monotone
f (0) = 0, f (x) Æ • as x Æ1
Asymptotic Stabilization of Inverted
Equilibrium of Lagrange Top
Theorem: The inverted equilibrium (Gi,0) is
asymptotically stable equilibrium of the closed
loop with a dense domain of attraction (S2 \
{Gh} x R2).
Asymptotic Stabilization of Inverted
Equilibrium of Lagrange Top
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Controller arises from control Liapunov function V : S 2 ¥ R 2 Æ R
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Controller is not globally defined: not defined at
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hanging configuration
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The controller is PD feedback: angular velocity
feedback and reduced attitude feedback
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Obtain a nearly global domain of attraction of the
inverted equilibrium; must exclude the hanging
configuration
Simulation Results
Simulation Results
Simulation Results
Asymptotic Stabilization of Arbitrary
Configuration of Spherical Pendulum
Assume c = 0. Consider the controller with feedback
of angular velocity and reduced attitude
Ê s 02 (G) ˆ
u x = mgr s Gy + J t k x y˙1 - (w x - k x y1 ) + f ' Á
˜y1
Ë 4 ¯
Ê s 02 (G) ˆ
u y = -mgr s Gx + J t k y y˙2 - (w y - k y y2 ) + f ' Á
˜ y2
Ë 4 ¯
where
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†
†
s 0 (G) = 1- G0T G , kx > 0, ky > 0
f :[0,1) Æ R,strictly monotone
f (0) = 0, f (x) Æ • as x Æ1
Asymptotic Stabilization of Arbitrary
Configuration of Spherical Pendulum
and
y1 = s 0 (G)(Gyo Gz - Gz0Gy )
y2 = s 0 (G)(GzoGx - Gx 0Gz )
Theorem: Let G0 e S2. The equilibrium (G0,0) is
asymptotically
stable equilibrium of the closed
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loop with a dense domain of attraction (S2 \
{-G0} x R3).
Asymptotic Stabilization of Arbitrary
Configuration of Spherical Pendulum
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Controller arises from control Liapunov function V : S 2 ¥ R 2 Æ R
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Controller is not globally defined
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Controller is PD feedback: angular velocity feedback and
reduced attitude feedback
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Obtain a nearly global domain of attraction of an arbitrary
configuration
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There is large literature on this problem
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Swing-up and switching controller structure widely studied
Usual models have singularities so that global analysis is not
possible
Simulation Results
Simulation Results
Simulation Results
Uncontrolled Asymmetric
3D Pendulum
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Assume the 3D pendulum is asymmetric, that is the principal
moments of inertia are distinct.
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Assume the pivot location and center of mass location are distinct
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Recall
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Total energy
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Jw˙ = Jw ¥ w + mgr ¥ G
G˙ = G ¥ w
1 T
w Jw - mgr T G is conserved
2
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Angular momentum component about the vertical axis
w T JG †is conserved
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Dynamics of asymmetric 3D pendulum evolve on S2 x R3
Equilibria of an Uncontrolled
Asymmetric 3D Pendulum
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Reduced equilibria of the 3D pendulum satisfy
w = 0 and r x G = 0
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There are two reduced equilibrium configurations
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r
Stable hanging equilibrium w = 0, G = G =
h r
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Unstable inverted equilibrium w = 0, G = Gi =
†
-r
r
Simulations of Uncontrolled Dynamics
of Asymmetric 3D Pendulum
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3D pendulum data
l J = diag[1, 2.8, 2] kg-m2, m = 1 kg, r = [0, 0, 1] m
l 3D pendulum is initially in horizontal plane
Simulation responses
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Motion appears to be chaotic
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Animation of Uncontrolled Dynamics
of Asymmetric 3D Pendulum
Asymptotic Stabilization of
Asymmetric 3D Pendulum
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Controlled asymmetric 3D pendulum
Jw˙ = Jw ¥ w + mgr ¥ G + u
G˙ = G ¥ w
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Use
feedback to asymptotically stabilize a reduced equilibrium of the
†
asymmetric 3D pendulum
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Angular velocity feedback
u : R3 Æ R3
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Angular velocity and reduced attitude feedback
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†
u : S2 ¥ R3 Æ R3
Existence of Smooth Stabilizing
Controllers for Asymmetric
3D Pendulum
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Global asymptotic stabilization of a reduced attitude
equilibrium of the asymmetric 3D pendulum is not possible
using any continuous feedback
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Reason: there is a topological impediment due to the
fact that the configuration space S2 is compact
What are possible closed loop domains of attraction?
l
The domain of attraction may be all of S2 x R3 except for
a set of Lebesgue measure zero; this is referred to as
almost global asymptotic stabilization.
Asymptotic Stabilization of Hanging
Equilibrium of Asymmetric
3D Pendulum
Consider the controller with feedback of angular
velocity
u x = -k x w x
u y = -k y w y
where kx > 0, ky > 0 .
†
Theorem: The hanging equilibrium (Gh,0) is
asymptotically stable equilibrium of the closed loop
with a dense domain of attraction (S2 x R3 ) \ M.
Asymptotic Stabilization of Hanging
Equilibrium
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Controller arises from control Liapunov function V : S 2 ¥ R 3 Æ R
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Controller is globally defined
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Angular velocity feedback only
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Nearly global domain of attraction of the hanging
equilibrium; must exclude the stable manifold of
the inverted equilibrium
†
Simulation Results
Simulation Results
Simulation Results
Asymptotic Stabilization of Arbitrary
Reduced Attitude Configuration
Consider the controller with feedback of angular
velocity and reduced attitude
È Ê s 02 (G) ˆ
˘
u = Íf ' Á
˜s 0 (G) - mgr ˙ ¥ G - k x w x - k yw y
4
Ë
¯
Î
˚
where
s 0 (G) = 1- G0T G , kx > 0, ky > 0
†
†
f :[0,1) Æ R, strictly monotone
f (0) = 0, f (x) Æ • as x Æ 1
Theorem: Let G0 e S2. The equilibrium (G0,0) is
†
asymptotically
stable equilibrium of the closed loop
with a dense domain of attraction (S2 \ {-G0} x R3).
Asymptotic Stabilization of Arbitrary
Reduced Attitude Configuration
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Controller arises from control Liapunov function
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Controller is not globally defined
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Feedback of angular velocity and reduced attitude
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Almost global domain of attraction of the specified
equilibrium
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Applies to asymptotic stabilization of hanging
equilibrium or to asymptotic stabilization of inverted
equilibrium
Simulation Result
Simulation Result
Computational Issues for
3D Pendulum
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Computations that preserve the geometry of the configuration space
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Use exponential representations for computations
that involve rotation matrices
Computations that preserve the geometry imposed by the presence
of conserved quantities
l Use integrators obtained from discrete variational
principles
Experimental Results for the
3D Pendulum
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Adaptive stabilization of the angular velocity
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Assume center of mass coincident with pivot point
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Tracking control of angular velocity only
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Adaptation to unknown inertia of pendulum
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Experimental results obtained from the TACT
Experimental Results for the
Asymmetric 3D Pendulum
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Asymptotic stabilization of hanging equilibrium
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Assume asymmetric 3D pendulum
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Assume center of mass is not coincident with pivot point
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Use feedback controller based on angular velocity feedback only
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Recent experimental results have been obtained from the TACT
Experimental Results
Experimental Results
Experimental Results
Experimental Results
Conclusions
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Propose 3D pendulum as benchmark system for
dynamics and control research
Study uncontrolled (free) dynamics of 3D pendulum
l Demonstrate existence of conserved quantities
l Multiple coupled rotational degrees of freedom
l Geometry of the 3D pendulum dynamics defined on
compact configuration space SO(3) or S2
l Suggest possibility of chaos in free dynamics
Study stabilization of 3D pendulum
l Asymmetric pendulum
l Stabilization of reduced hanging equilibrium using
angular velocity feedback only
l Stabilization of arbitrary equilibrium using angular
velocity and reduced attitude feedback
Conclusions
Axially symmetric pendulum
l Stabilization of Lagrange top
l Stabilization of spherical pendulum
Study computational dynamics
l Preserve geometry of configuration space
l Preserve geometry of any conserved quantities
Experimental studies using TACT
l Combined theoretical and experimental research
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Conclusions
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What are the novel features of this research?
l “New” 3D pendulum
l Importance of the geometry of the 3D pendulum
dynamics
l New results for free dynamics
l New stabilization results
l Liapunov based controllers defined on S2
l Geometry of closed loop domains of attraction
l Avoid traditional use of controller switching
l Importance of computational dynamics
l Combined theoretical and experimental research