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Research Papers

Safe Control of Hopping in Uneven Terrain

[+] Author and Article Information
Brian Howley

 Lockheed Martin Space Systems Company, Sunnyvale, CA 94089

Mark Cutkosky

 Stanford University, Stanford, CA 94305

J. Dyn. Sys., Meas., Control 131(1), 011012 (Dec 08, 2008) (11 pages) doi:10.1115/1.3023133 History: Received March 27, 2008; Revised July 25, 2008; Published December 08, 2008

Using a classic problem from robotics of a vertical hopping machine, we demonstrate an approach for investigating the safety of a hybrid discrete/continuous dynamic system operating in an uncertain environment. The challenges imposed by the environment are expressed in terms of constraints imposed in the phase space of the system as it undergoes periodic motion. The approach is demonstrated first with a hopper that has state feedback to govern the timing of thrust and subsequently for a timer-based hopper. The latter case increases the dimensionality of the problem and must be treated numerically. However, the use of a multiresolution surface representation of the feasible regions in state space reduces the computational burden of the approach.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of a vertical hopper. The hopper dynamics are defined by the height, y, and velocity, ẏ, with respect to a reference and the height, h. Parameters affecting hopper dynamics include undeflected leg length, l, spring and damping constants, and maximum thrust capability.

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Figure 2

Hopper flight and contact phases of motion. To accommodate a sudden change in height above ground flight is divided into ascent , step, and descent modes. Furthermore, the contact phase is divided into contact and liftoff . The liftoff mode fixes the value of the leg thrust. If leg thrust is zero hopper motion is ballistic. If leg thrust is umax the motion remains in contact .

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Figure 3

Example results for vertical hopper under feedback control. (a) shows the boundaries in state space for different dynamic modes. The singularity avoidance (SA) region is shown with dotted lines. The leg compression constraint, ŷ<ŷmin=−0.5, is the shaded area in the contact region. (b) shows the curves for forward and backward time propagation at û=ûmax. (c) shows the results for one iteration of the algorithm. (d) shows convergence towards a steady state solution. The invariant safe set is the unshaded region bounded by the steady state curves.

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Figure 4

Maximum tolerance to terrain variability as a function of stiffness and damping. For near zero damping maximum ruggedness is achieved at high spring stiffness. But at even small levels of damping, moderate to light spring stiffness is better. Optimal damping increases with decreasing spring stiffness.

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Figure 5

Hopper dynamic modes under open loop control. The contact phase is divided into three new modes: compression , thrust , and expansion . The transition between modes is controlled by the timer state variable, ŝ. The timer resets to zero upon reaching ŝ=ŝmax. Thrust is applied when in ground contact and when 0⩽ŝ⩽ŝthrust.

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Figure 6

Initial boundary for compression mode: (a) compression mode initial and exit condition manifolds; (b) initial boundary after propagating from initial to exit conditions and merging sub-boundaries; (c) propagated constraint sub-boundary; and (d) initial boundary after resect and merge (note change of scale)

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Figure 7

Backward chaining operations on mode Q. The safe set boundary for mode Q begins with initial and exit condition sub-boundaries, Xinit(Q+) and Xexit(Q+), respectively. The intersection with with successor mode initial conditions, XinitS−, forms a new set of exit conditions, Xexit(Q−). Backward time propagation from the resulting border forms a new initial condition sub-boundary, Xinit(Q−).

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Figure 8

First iteration of the backward chaining sequence for stair climbing. The sequence starts with step mode IC and EC sub-boundaries in the upper left panel and propagates backward in time through ascent , expansion , thrust , compression , and descent modes. Only initial and exit condition sub-boundaries are shown for clarity.

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Figure 9

Steady state step mode exit condition sub-boundary for stair climbing. State space trajectories originating within this sub-boundary will remain safe and return within this boundary. The hopper will eventually settle into a steady state cycle that alternates between the two stable fixed points exhibiting a limped gait behavior.

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