Marcotte Dynamic Stabilization of the Inverted Pendulum

Dynamic Stabilization of the Inverted Pendulum

C. Marcotte,1 B. Suri,1 J. J. Aguiliar,1 and G. Lee1

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332,USA

(Dated: 16 December 2011)

The inverted pendulum is a canonical problem in both Nonlinear Dynamics1 and ControlTheory2,3. In this article, the phenomenon of dynamic stabilization of the vertically driveninverted pendulum is investigated experimentally and numerically. We resolve the first sta-bilizing boundary in driving parameter space, as well as investigate the effects of frictionaldamping on the dynamics of the pendulum. We finish our investigations with the dynamicsof the Double Inverted Pendulum.

I. INTRODUCTION

Originally investigated by P. L. Kapitza4, andlater, Kalmus5, the inverted pendulum has beenshown to have a stable inverted state when it’scentral pivot is subject to high-frequency verti-cal displacement under a variety of waveforms.6

On the subject of dynamical stability, Kapitzahad this to say:

[T]he striking and instructivephenomenon of dynamical stabilityof the turned pendulum not onlyentered no contemporary handbookon mechanics but is also nearly un-known to the wide circle of special-ists... ...[The phenomenon of dy-namical stability is] not less strik-ing than the spinning top and asinstructive.7

We begin with a short investigation of the the-ory of dynamic stabilization before continu-ing to the experimental Methods and Discus-sions.

II. THEORY

The driven inverted pendulum possesses a ro-tational symmetry about the vertical axis; tosimplify the analysis, we look at solely planaroscillations. Such a system is replicated exper-imentally. For an inverted pendulum of massm and length `, affixed at a pivot at position ywith angular displacement from the vertical up

FIG. 1. Diagram of the theoretical model of theinverted pendulum.

position given by θ, the Lagrangian is given byEq. 1.

L = m2 (`2θ2 + y2 + 2`yθ sin θ)−mg(y + ` cos θ)

(1)

Which gives the equation of motion, in scaled,dimensionless parameters α = g/`ω2, β = b/`and τ = ωt,

0 = θ + (βf(τ)− α) sin θ (2)

Such that ∂2τy(t) = bf(τ), and θ is understoodto be differentiated with respect to τ . We in-troduce damping to the model through the ad-

dition of a frictional term to Eq. 2, whereˆθ

1

is the sign of θ and γ is a constant frictioncoefficient.8

0 = θ + γˆθ + (βf(τ)− α) sin θ (3)

Clearly, a system described by Eq. 2 possessestwo equilibria: (θ∗, θ∗)+ = (0, 0) corresponding

to the upward vertical and (θ∗, θ∗)− = (π, 0) forthe downward vertical. Linearizing around theequilibria and making the perturbative substi-tution η± = θ∗± + δθ±, we obtain the Mathieuequation9.

δθ± ∓ (βf(τ)− α)δθ± = 0 (4)

The linear stability analysis confirms our sus-picions: the stability of the (±)-states is de-termined by the pre-factor (α − βf(τ)), withthe (+)-state losing stability as β → 0. It

has been well documented10 that Eq. 4, gov-erning the stability of the inverted state, im-plies the existence of a series of bifurcations and“resurrections” as the driving amplitude is var-ied for fixed driving frequency. With this inmind, limited by the equipment, we resolvedonly the lower boundary of the first stable re-gion over a range of values for the relevant pa-rameters.For completeness, we will here include the gov-erning equations for a series-connected dou-ble pendulum with the first pivot driven sinu-soidally. Solved for using the same Lagrange-Euler formalism as in the single pendulum case,we determine two second-order, nonlinear, cou-pled differential equations in the variables θ1and θ2, which are the angular displacements ofthe inner and outer arms of the pendulum, re-spectively.

θ1 =− m2(L1θ21 sin(2θ1−2θ2)+2L2θ

22 sin(θ1−θ2))+g((2m1+m2) sin(θ1)+m2 sin(θ1−2θ2))−y((2m1+m2) sin(θ1)+m2 sin(θ1−2θ2))

2L1(m2sin(θ1−θ2)2+m1)

θ2 =L2m2θ

22 sin(2θ1−2θ2) +(m1+m2)(2L1θ

21 sin(θ1−θ2)+g(sin(2θ1−θ2)−sin(θ2)))+y(m1+m2)(sin(θ2)−sin(2θ1−θ2))

2L2(m2sin(θ1−θ2)2+m1)(5)

III. METHODS

A. The Pendulums

There are three iterations of the pendulum rep-resented in this article. The first two are singlependulums, and the third is a series-connecteddouble pendulum.

1. The First

The first pendulum was constructed from alu-minum, with an effective length of 6.88 cm.It consisted of a central rectangular rod at-tached to a near-cube component with two em-bedded high-end skateboard bearings which ro-tated about an axle supported on the farthestends. Unfortunately, one of the bearings ex-hibited “graininess” and introduced a large θ-dependent frictional term, which was especially

FIG. 2. The first iteration of the pendulum, witheffective length 6.88 cm and constructed from alu-minum.

pronounced in the vicinity of the fixed point (+)at low angular speeds. This added measurablenoise to the measurement of the ring-down, andcreated small local energy minima causing thependulum to settle into different angular posi-tions when at rest, which manifested as a “walk-

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FIG. 3. The second iteration of the pendulum, witheffective length 3.13 cm and constructed from alu-minum. It is this iteration which is used for theanalysis of the system.

ing” behavior when driven.

2. The Second

The second pendulum is a modification of theoriginal. Still constructed of aluminum, oneof the vertical supports is removed, as wellas one of the ball-bearings. Additionally, thearm of the pendulum was drastically short-ened. The result is a noticeable improvementin the smoothness of traversal in the vicinity ofthe fixed point at low angular speeds, and thenear elimination of local minima of the energyaround the fixed point due to θ-dependent fric-tion. Further, with less mass and a shorter arm,the pendulum is easier to excite, as the ampli-tude of the driving function is inversely propor-tional to the pendulum length: β ∝ `−1.

3. The Third

The double pendulum was constructed fromLego parts in an asymmetric way such thatthe pendulums would not collide. This was ac-complished by extending each pendulum furtherfrom the support than the previous. In this way,the planes described by the “whirling modes”of each separate pendulum do not intersect inspace. Unfortunately, due to the flexibility ofthe axles connecting the pendulums to the sup-

FIG. 4. The final pendulum investigated: the dou-ble pendulum. The support is aluminum, and theLego construction has an effective length of 3.2 cmvertically, while extending approximately 2 cm hor-izontally from the support.

port and each other, when driven the pendu-lums exhibit large oscillations out of their re-spective planes. This enables collision of thependulums, and represents a strongly damped,dissipative mode in addition to the idealized dy-namics. The support is the same as before, con-structed from aluminum, but the pendulum isaffixed by clamps under tension to it.

B. Additional Materials

In addition to the menagerie of pendulums, theuse of a shaking table was required, which wasoutfitted with an accelerometer. The output ofthe accelerometer was interpreted and displayedon an oscilloscope to measure the peak-to-peakvoltage (a measure of the amplitude of the driv-ing function). Further, a function generator andamplifier were used to supply the shaking ta-ble with a displacement to attain. For the pur-poses of tracking the pendulum as the systemevolves, electrical tape (in the canonical black),whiteout fluid, and small white plastic sphereswere introduced to create regions of arbitrarilyhigh contrast at both the central pivot and theapproximate location of the center of mass inthe pendulum. Finally, two high-speed cameraswere used, one Gray Point for real-time trackingand the other RedLake for offline higher frame-rate tracking, in Labview and MATLAB.

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C. Collection Methods

To resolve the boundary of the first stability re-gion of the inverted state, in essence, we arespanning a two-dimensional parameter spaceand measuring the binary value: is the invertedstate stable? Practically, we chose a driving fre-quency on the lower end of the range of avail-able driving frequencies, and with the pendu-lum slightly displaced from the inverted state,slowly increased the amplitude of the drivingwaveform until the fixed point becomes locallyattractive. For the tracking of the dynamics,high-contrast agents (electrical tape and white-out or white plastic spheres) and two high-speedcameras were used. The real-time tracking al-gorithm employed in Labview requires severalintervening actions from the participants. Thefirst is the manual selection of the trackingmarkers, followed by setting a threshold to sepa-

FIG. 5. Photograph of the second and third pendu-lums, clarifying the use of high contrast markers.

rate the marker highlights from the background.Finally, the program searches a square of pointsaround the initially selected point and finds thecenter of the pixels which have brightness val-ues above the threshold. This computed centerpoint is then used as the search seed for the nextframe, and the process repeats itself. The offlinetracking done in MATLAB uses the same algo-rithm, with a more robust threshold and largersquare search size. One issue with this searchmethod is that when obscured, tracking pointswould become lost or (in the double pendulumcase) overlap and merge. To circumvent theseissues, and because the algorithm does not in-terpolate or predict and does not use global in-formation to find the tracking point, we had tomanually re-select the tracking points and con-tinue the search.

IV. RESULTS

We begin with the efforts to resolve the stabil-ity boundary of the single inverted pendulumstate. The results are plotted in Fig. 6. Theexperimental results are compared to the the-oretical predictions in Fig. 7. We include theresults of Blackburn, et al, for comparison inFig. 8.11

15 20 25 30 35 40 454

5

6

7

8

9

10

11

12Experimental Stability Mapping

Forcing Frequency (Hz)

Forc

ing A

mplit

ude (

mm

)

FIG. 6. Experimentally resolved stability boundaryof the single inverted pendulum state θ+.

What follows next is the result of trackingdata, and the reconstructed phase portraitsthereof. Also included are phase portraits gen-

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FIG. 7. Experimentally resolved stability boundaryof the single inverted pendulum state θ+. Here, thehorizontal axis is scaled such that ε = ω2

0A/g, andΩ = ω/ω0, where g is the acceleration due to gravityand ω0 = 2π · 1.9 Hz is the natural frequency of thependulum, determined from the ring-down trackingdata. The experimental data is in white (error barsomitted for clarity), and the theory predicts stabil-ity in the green region.

FIG. 8. Results reproduced from Blackburn etal, for the theoretical predictions of the stabilityboundary.11

erated from numerical models of the same sys-tem. In short, Fig. 10 should be comparedto Fig. 11, and Fig. 12 should be comparedto Fig. 13. Note, however, that these phaseportraits are, in fact, not phase portraits: the

dynamics clearly intersect. These are projec-tions of the space (θ, θ, φy) onto (θ, θ), wherey(t) = Aeiφyt.

FIG. 9. Raw tracking time-series (xn, yn) for thesingle pendulum, in absolute pixels on the left, withthe effect of the driven pivot removed.

To transform the (xn, yn, tn) time-series to a(θn, tn) time-series is simple geometry: θn =

tan−1(xn/yn). To construct θn is less triv-ial. Since the temporal spacing is not uni-form (∀i,j 6=i : ti − ti−1 6= tj − tj−1), spectralmethods would require interpolation betweenthe existing data points. This interpolation re-lies on assumptions about the data set whichcould introduce systematic errors into the eval-uation of θ, negating the benefit of comput-ing the derivative spectrally. For this reason,and computational clarity, we chose a nearest-neighbor coupling of the differential operator:θn = (tn − tn−1)−1(θn − θn−1).

Next we present the determination of the damp-ing parameter which corresponds to a constantfrictional damping. By fitting a quadratic func-tion to the profile of the θn series, we can deter-mine if the damping is frictional (if the profileis linear) or viscous (if the profile is exponen-tial).

Finally, we present the results of our investi-gations concerning the frequency of small os-cillation about the stable inverted state forvaried driving frequencies and driving ampli-tudes.

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FIG. 10. Phase portrait constructed from thetracking of the single pendulum at a driving fre-quency of 25 Hz. The units on the horizontal andvertical axes are degrees and degrees per second,respectively.

FIG. 11. Phase portrait constructed from simula-tion of the single pendulum at a driving frequencyof 25 Hz. The units on the horizontal and verticalaxes are radians/π and radians per second, respec-tively.

V. DISCUSSION

As concisely put in Fig. 7, over the limitedrange of driving parameters available to us andthe equipment, there is good agreement with

FIG. 12. Time-series representation of θ1,n, θ2,n,for the stable-up (0, 0) state under sinusoidal driv-ing of the central pivot, determined experimen-tally.

FIG. 13. Verification of the upward-stable state forthe series-connected double pendulum under sinu-soidal forcing of the central pivot by simulation. Inthe upper left is a image representation of the (0, 0)-state, in the upper right is the extremal position ofthe outer pendulum arm, and the bottom two plotsare phase portraits for the inner pendulum (left)and outer pendulum (right).

theory for the boundary of the first stable re-gion of the inverted state. Further, there isgood qualitative agreement in the phase por-traits generated from experimental data and nu-merical simulation, both for the single and dou-

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FIG. 14. Phase portrait constructed from thetracking of the double pendulum at a driving fre-quency of 24 Hz exhibiting the dynamics of a “flip-ping” mode. The units on the horizontal and verti-cal axes are degrees and degrees per second, respec-tively.

FIG. 15. Time-series representation of the “flip-ping” mode observed experimentally in the dou-ble pendulum, included for clarity.

ble pendulums.Finally, we turn our attention to the “flipping”mode depicted in the phase portrait Fig. 14,and as a time-series in Fig. 15. These aretwo anti-symmetric states: (θ1, θ2) = (0, π) and(π, 0). In this regime, we suspect, the smallangle approximation applies to the dynamical

FIG. 16. The results of the “ring down” track-ing data, and a quadratic fit to the profile. Thefitting-function is a polynomial in t : f(t; a, b, c) =at2 + bt + c. The fit converges with a RMSEof 0.006664, with the pre factor on the quadraticterm O(b/10), indicating a primary contribution todamping which is constant in time, or frictional.The pre-factors are a = 0.0083 ± 0.000687, 95%,b = −0.198±0.0054, 95%, and c = 1.28±0.009, 95%.

equations Eq. 5, and small angle deviancesof one pendulum may be considered as inertialrestoring forces to the other. This explains whythe symmetric states (0, 0), (π, π) do not ex-hibit this phenomenon. As a result, the outerpendulum exhibits nearly independent dynam-ics. Intuitively, the gauge freedom of the po-tential energy implies that for these symmetrypreserving states the position of the inner pen-dulum simply defines a higher or lower potentiallevel of the outer pendulum for the θ1 = 0 andθ1 = π states, respectively. However, furthertheoretical work is needed to verify this hypoth-esis.

VI. CONCLUSION

In conclusion, we were able to stabilize boththe single pendulum and the double pendulumusing only vertical sinusoidal forcing of the cen-tral pivot. We were able to track the dynam-ics with adequate spatial and excessive tempo-ral resolution and compare their shape to the

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FIG. 17. By fitting over the first oscillatory periodwith a sine wave and using the previously deter-mined (see Fig. 16) semi-linear profile to determinethe form of the damping term in Eq. 3, we maydetermine the value of the damping coefficient γ.From the fit initial condition and the damping co-efficient, the model is integrated in time using anexplicit Euler scheme to verify the predictability ofthe dynamics.

simulation results with good qualitative agree-ment. We resolved the first occurrence of a sta-bility boundary for the single pendulum in theinverted state. Additionally, we were able toobserve positive correlation of the frequency ofsmall oscillations about the inverted state forthe single pendulum with driving amplitude andfrequency. Finally, we observed separable dy-namics in a set of antisymmetric states for thedouble pendulum.

REFERENCES

1S. Strogatz, Nonlinear dynamics and chaos: withapplications to physics, biology, chemistry, andengineering (Perseus Books, 1994).

2D. Liberzon, Switching in Systems and Control(Springer, 2003).

3Franklin, Feedback control of dynamic systems(Prentice Hall, 2005).

4P. Kapitza, “Dynamical stability of a pendulumwhen its point of suspension vibrates, and pendu-lum with a vibrating suspension,” Collected Pa-pers of PL Kapitza 2, 714–737 (1965).

FIG. 18. The results of the investigation intowhether the frequency of small oscillations aboutthe inverted state depended on the driving fre-quency and amplitude. We note a tenuous positivecorrelation between the amplitude of the accelera-tion and the frequency of small oscillations aboutthe stable inverted state, (+) for each driving fre-quency, and across increasing driving frequencies.

5H. Kalmus, “The inverted pendulum,” AmericanJournal of Physics 38, 874 (1970).

6E. Yorke, “Square-wave model for a pendulumwith oscillating suspension,” American Journal ofPhysics 46, 285 (1978).

7P. L. Kapitza, Collected papers of P. L. Kapitza,edited by D. T. Haar, Vol. 2 (Pergamon London,1976) pp. 714, 726.

8A. Marchewka, D. Abbott, and R. Beichner,“Oscillator damped by a constant-magnitude fric-tion force,” American journal of physics 72, 477(2004).

9M. Bartuccelli, G. Gentile, and K. Georgiou, “Onthe dynamics of a vertically driven damped pla-nar pendulum,” Proceedings of the Royal Societyof London. Series A: Mathematical, Physical andEngineering Sciences 457, 3007 (2001).

10S. Kim and B. Hu, “Bifurcations and transitionsto chaos in an inverted pendulum,” Physical Re-view E 58, 3028 (1998).

11J. Blackburn, H. Smith, and N. Grønbech-Jensen, “Stability and hopf bifurcations in an in-verted pendulum,” American journal of physics60, 903–908 (1992).

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