Research Papers

Steinkamp's Toy Can Hop 100 Times But Can't Stand Up

[+] Author and Article Information
Gregg Stiesberg

Department of Physics,
Cornell University,
Ithaca, NY 14850
e-mail: grs26@cornell.edu

Tim van Oijen

Civil Engineering and Geosciences,
Delft University of Technology,
Delft 2600 AA, The Netherlands
e-mail: T.P.vanOijen@tudelft.nl

Andy Ruina

Departmen t of Mechanical Engineering,
Cornell University,
Ithaca, NY 14850
e-mail: ruina@cornell.edu

1Corresponding author.

Manuscript received September 8, 2016; final manuscript received November 8, 2016; published online January 13, 2017. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 9(1), 011017 (Jan 13, 2017) (13 pages) Paper No: JMR-16-1265; doi: 10.1115/1.4035337 History: Received September 08, 2016; Revised November 08, 2016

We have experimented with and simulated Steinkamp's passive-dynamic hopper. This hopper cannot stand up (it is statically unstable), yet it can hop the length of a 5 m 0.079 rad sloped ramp, with n100 hops. Because, for an unstable periodic motion, a perturbation Δx0 grows exponentially with the number of steps (ΔxnΔx0×λn), where λ is the system eigenvalue with largest magnitude, one expects that if λ>1 that the amplification after 100 steps, λ100, would be large enough to cause robot failure. So, the experiments seem to indicate that the largest eigenvalue magnitude of the linearized return map is less than one, and the hopper is dynamically stable. However, two independent simulations show more subtlety. Both simulations correctly predict the period of the basic motion, the kinematic details, and the existence of the experimentally observed period 11 solutions. However, both simulations also predict that the hopper is slightly unstable (|λ|max>1). This theoretically predicted instability superficially contradicts the experimental observation of 100 hops. Nor do the simulations suggest a stable attractor near the periodic motion. Instead, the conflict between the linearized stability analysis and the experiments seems to be resolved by the details of the launch: a simulation of the hand-holding during launch suggests that experienced launchers use the stability of the loosely held hopper to find a motion that is almost on the barely unstable limit cycle of the free device.

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Fig. 1

The Steinkamp hopper: (a) the entire device, (b) close up of body (compare with Fig.2(a)), (c) the leaf spring, and (d) the feet (with motion-capture reflective marker tape on one). For more details, see Ref. [1].

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Fig. 2

Schematic of the simulation model. (a) The rigid body (gray) and the rigid leg (dark outline with white fill) are connected by a leaf spring (black). The “leg at rest” configuration (striped fill) corresponds to the leg configuration shown in Fig.1(b). The leaf spring allows displacement normal to its length and also rotation (the rotation is never so large as to cause leg collision with the counterweights on the body). The net effect is allowance of leg rotation (leg swing) about the nominal hip and of leg axial motion (body bounce). Horizontal motion of the leg, relative to the body, at the nominal hip, is effectively constrained to zero by the extension stiffness of the leaf spring. (b) In the 2D modeling, the body and leg each have two position coordinates and an angle. The constraint from leaf-spring's inextensibility is modeled using a pin in a slot. In one of our simulation models, the slot is in the body (shown), and in the other, the slot is in the leg. The leaf spring is modeled as a coupled torsion–extension viscoelastic spring in which torque and force both depend on displacement and rotation, with the elastic part determined by classical beam theory.

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Fig. 3

One step of hopper motion. The photo is rotated slightly counterclockwise so that the actual slope, down and to the right, appears as horizontal. The motion is from left to right. The natural orientation of the device, while hopping, is rotated ≈10 deg counterclockwise compared to schematic views in Figs. 1(b) and 2(a). One hop cycle is shown from simulation (discussed at length below) and from slow motion video. (a) At liftoff, the backward force on the foot from the ground is relieved, and the body moves upward; (b) as the device moves forward and upward, the leg swings forward; (c) near the apex of the hop, the leg movement reverses direction (the leg comes close to, but does not collide with the body); (d) while dropping in height, just before the foot collides with the floor, the leg is still swinging backward relative to both body and ground (swing leg retraction [30,31]); and (e) in contact with the floor, the leg compresses and pushes the body upward again (video stills courtesy of Steinkamp).

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Fig. 4

Hand-held launch. (a) The tail of the device is held gently above the back of the body (Steinkamp's right hand) for the first ≈10 to 15 hops, then released. See text for details (photo background darkened for clarity, the slope is down and to the right but appears as nearly level due to camera perspective); (b) shock-absorber-attached model of launch. In simulations of launch, we model the right-hand holding as a shock absorber attached between the point (xm, ym) on the tail, and where the hand is an imaginary reference point moving at a constant launch speed vl.

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Fig. 5

Foot position versus distance, measured and simulated. (a) A magnified view of the path of a point on the foot from motion-capture data and (c) from simulation. Note the small loop made by the foot as it pivots and lifts off. The vertical scale is exaggerated to enhance the details of the motion. Relative angle versus time, measured and simulated. (b) The relative angle between the body and leg from motion-capture data and (d) from simulation.

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Fig. 6

Period ∼11 oscillations. (a) In motion-capture data, the modulation of foot oscillations repeats approximately, but not exactly, every 11 hops. (b) As one might extrapolate from the argument of the totter eigenvalue (see text), the nonlinear simulation shows a variation in hopping amplitude that repeats about every 11 hops. (a) Motion capture—foot height versus distance hopped and (b) simulation—foot height versus distance hopped.

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Fig. 7

(a) Nonslip rolling constraint. Rolling without slipping is enforced by the constraint r˙c=0, where rc=rcom+rc/com. (b) Geometric parameters. There are eight geometric parameters, measured in the spring-relaxed configuration using a coordinate system at the hinge and oriented with the slot: two COM positions (4), foot position (2), foot radius (1), and ground slope (1); 4 + 2 + 1 + 1 = 8. (c) Mechanical parameters. There are 11 mechanical parameters: masses (2), polar inertias (2), spring constants (3), damping constants (3), and gravity (1); 2 + 2 + 3 + 3 + 1 = 11. See text and Appendix A online under the “Supplemental Data” tab for this paper on the ASME Digital Collection.

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Fig. 8

Comparison of two models. (a) The model implemented by Stiesberg (GS), with the slot fixed in the body, and (b) the model implemented by van Oijen (TvO), with the slot fixed in the leg. (c) The difference between simulations of van Oijen and Stiesberg Δ(δ)=|qfL−qfB|, normalized by the magnitude δ of the initial state perturbation δ=|qi−q*|, is shown as a function of perturbation magnitude δ. Linear convergence is seen as δ decreases to δ≈1×10−5, at which point round-off error begins to dominate. Note that the minimum difference between the solutions Δ/δ≈10−5, at  log10δ≈−5.5, corresponds to an actual difference Δ of ≈1×10−10 because of the normalization by δ; this normalization is also responsible for the slope ≈−1 on the left of the graph which actually indicates a constant, not increasing, error with decreasing step size. For all of these calculations, an integration time step was chosen such that the errors above dominate the integration method errors.

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Fig. 9

Simulation eigenvalues. X's mark the complex eigenvalues on the complex plane. The unit circle is also shown. Parameters used are those that best fit the physical device and its motion. Only the speed eigenvalue is (barely) outside the unit circle; it is positive and real. The totter eigenvalues are slightly stable complex conjugate pair, and the swing eigenvalues are a quite stable (far inside the unit circle) complex conjugate pair.

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Fig. 10

Optimization of maximum simulation eigenvalue. The eigenvalues corresponding to the initial parameter set and the last accepted parameter set are marked by arrows labeled “start” and “end,” respectively. The initial parameter set was intentionally chosen to destabilize the totter modes. The optimization succeeds in stabilizing the totter modes, but not the speed mode, which remains very close to λ = 1 for the entire optimization run. All the iterations in this optimization run are shown: the successes that reduce the maximum eigenvalue and the failures that sometimes drastically increase the maximum eigenvalue. The problem is that the eigenvalues for the swing eigenmodes become real, with one of them growing quickly along the negative real axis and thus obviating reduction of the speed mode eigenvalues.

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Fig. 11

Maximum eigenvalue during simulated launch. The maximum eigenvalue magnitude for the periodic motion of the held hopper (the modeled launch condition) is shown as a function of spring and damping parameters kl,cl, with a constant launch speed vl≈0.551, equal to the free hopping map fixed point speed. This launch map has a stable fixed point over the majority of the parameter space shown. The boundary of stability, the |λ|=1 contour, overlaps the interval of the vertical axis shown here for cl>≈0.03; on this interval, numerics and reasoning agree that the maximum eigenvalue approaches exactly 1 as kl approaches 0 (see text). The coordinate of the attachment point in the body-fixed coordinates (see Fig. 4(b)) is (xm,ym)=(−0.8,1.0)*ℓ. The stability depends on the attachment coordinate ym; with too low a grip, the sliding ring contact does not adequately couple to the totter mode. With too high a grip, the dashpot does not adequately couple to the hopper's speed mode.

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Fig. 12

Hopping distance as a function of launch speed. The hopper is launched from a stable periodic state of the shock-absorber held system. For each different speed, there is a different perturbation from the free periodic motion. Using this launch model, simulations predict a hopping distance greater than 4 m for a range of launch speeds 0.544<vl<0.609 ; the magnitude of this interval is 0.065, somewhat but not substantially larger than the interval for a simulation with a bumpy slope (compared to Fig. 17(b) in Appendix C under the “Supplemental Data” tab for this paper on the ASME Digital Collection). The spring and damping coefficients are kl=0.5  and cl=0.5. The fixed point of the free hopping map has a speed ν*≈0.551. Note that even launching at the free-hopper limit-cycle speed, the distance to falling is not infinite because the held hopper has a different fixed point than the free hopper. Launch force as function of launch speed. The average launching force is a smooth monotonically decreasing function of the launch speed and crosses zero near the unstable free hopping map fixed point speed ν*. The average forces exerted on the device during this type of launch are similar to that of a penny resting on your finger, about 0.03 N. For launch speeds less than ν*, the instability would cause the robot to decelerate. Therefore, a positive force is needed to maintain constant speed. Conversely, at launch speeds greater than ν*, the instability would cause the robot to accelerate; therefore, a negative force is needed.

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Fig. 13

Launch force as a function of launch speed—a closer look. A magnified view of the force shows that it does not vanish at the free limit cycle speed ν*. This is consistent with the fact that the limit cycle at launch is not the same as the free limit cycle and indeed never can be the same for this launch technique.




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