Research Papers

Design for Control of Wheeled Inverted Pendulum Platforms

[+] Author and Article Information
Hari Vasudevan

Department of Mechanical Engineering and
Materials Science,
Yale University,
New Haven, CT 06511
e-mail: hari.vasudevan@yale.edu

Aaron M. Dollar

Assistant Professor
Department of Mechanical Engineering and Materials Science,
Yale University,
New Haven, CT 06511
e-mail: aaron.dollar@yale.edu

John B. Morrell

Assistant Professor
Department of Mechanical Engineering and
Materials Science,
Yale University,
New Haven, CT 06511
e-mail: john.morrell@alum.mit.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received August 15, 2013; final manuscript received November 24, 2014; published online March 11, 2015. Assoc. Editor: Xianmin Zhang.

J. Mechanisms Robotics 7(4), 041005 (Nov 01, 2015) (12 pages) Paper No: JMR-13-1162; doi: 10.1115/1.4029401 History: Received August 15, 2013; Revised November 24, 2014; Online March 11, 2015

In this paper, we study five aspects of design for wheeled inverted pendulum (WIP) platforms with the aim of understanding the effect of design choices on the balancing performance. First, we demonstrate analytically and experimentally the effect of soft visco-elastic tires on a WIP showing that the use of soft tires enhances balancing performance. Next, we study the effect of pitch rate and wheel velocity filters on WIP performance and make suggestions for design of filters. We then describe a self-tuning limit cycle compensation algorithm and experimentally verify its operation. Subsequently, we present an analytical simulation to study the effects of torque and velocity control of WIP motors and describe the tradeoffs between the control methodologies in various application scenarios. Finally, to understand if motor gearing can be an efficient alternative to bigger and more expensive direct drive motors, we analyze the effect of motor gearing on WIP dynamics. Our aim is to describe electromechanical design tradeoffs appropriately, so a WIP can be designed and constructed with minimal iterative experimentation.

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Fig. 2

(a) Charlie—balancing and (b) system architecture

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Fig. 1

Block diagram of system architecture with aspects of WIP design addressed in this paper highlighted with arrows

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Fig. 8

Simulated output of WIP response to disturbance of 0.0873 rad/s in pitch rate under the following filter configurations. (a) and (d) Wheel velocity filter fc = 5.0 Hz and pitch rate filter fc = 5.0 Hz; (b) and (e) wheel velocity filter fc = 5.0 Hz and pitch rate filter fc = 50.0 Hz; (c) and (f) wheel velocity filter fc = 50.0 Hz and pitch rate filter fc = 5.0 Hz. Note the instability of the response in plots (c) and (f).

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Fig. 3

The effect of tire damping (Bw) and drive train damping (Bwp) on pole-placement gains. Note that the X-axis represents the damping in both Bw and Bwp in all figures. (a) Comparison of pitch gains, (b) comparison of wheel position gains, (c) comparison of pitch rate gains, and (d) comparison of wheel velocity gains. The effect of tire–ground damping in simulation with an initial velocity of 0.1745 rad/s in pitch rate (e) Bw = 0 N m s/rad and (f) Bw = 0.1 N m s/rad.

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Fig. 4

Tire testing setup: (a) two tires were tested, 84 mm diameter on left and 90 mm diameter on right; (b) steel and aluminum rods of equal dimensions weighing 0.912 kg and 2.220 kg were loaded on the cart; and (c) cart instrumented with optical encoder to measure position and velocity

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Fig. 5

Rolling resistance tests. (a) No load on cart, (b) cart loaded with 0.912 kg aluminum rod, and (c) cart loaded with 2.220 kg steel rod.

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Fig. 6

Phase plots showing relative stability—(a) pendulum pitch versus pitch rate with hard tires, (b) pendulum pitch versus pitch rate with soft tires, (c) wheel position versus velocity with hard tires, and (d) wheel position versus velocity with soft tires

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Fig. 9

(a)–(f) Phase plots with various filter configurations, filter cutoffs are indicated: (a) and (d) phase plots with equal cut off frequencies. (b), (e) and (c), (f) Phase plots with different cutoff frequencies.

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Fig. 13

(a) Variation of pitch gain with motor constants, (b) variation of wheel position gain with motor constants, (c) variation of pitch rate gain with motor constants, and (d) variation of wheel velocity gain with motor constants. Note: the wheel velocity gain is the only gain that reduces the stability margin of the voltage controlled WIP.

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Fig. 14

(a) Pitch gain variation with gearing ratio, (b) wheel position gain with gearing ratio, (c) pitch rate gain with gearing ratio, (d) wheel velocity gain with gearing ratio, and (e) and (f) simulation of WIP response to a disturbance of 0.1745 rad/s in pitch rate with gearing ratios N = 50 and N = 200

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Fig. 10

Interpretation of the behavior of the balancing machine based on the sign of mechanical power terms, quadrants III and IV represent stabilizing conditions

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Fig. 11

Two types of limit cycle behavior depending on the magnitude of the compensation term, Vfc. From 0 to 40 s, the plots display limit cycles generates due to an underestimation of Vfc while from 40 to 100 s limit cycles due to and over-estimation of Vfc are displayed. (a) Wheel and pendulum power products. The negative spikes in Pw and positive spikes of Pp between 0 and 40 s indicate deviation quadrants IV and III, and (b) pitch and wheel positions.

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Fig. 12

(a) Friction compensation algorithm and (b) phase plot of operation of compensation algorithm. Time is encoded in color with red representing t = 0 s and blending into dark gray at t = 35 s. Note the reduction in limit cycles indicated by small central dark gray orbit.



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