Damping-induced self-recovery phenomenon in mechanical systems with - PowerPoint PPT Presentation

Damping-induced self-recovery phenomenon in mechanical systems with an unactuated cycle variable Dong Eui Chang Applied Mathematics, University of Waterloo 12 April 2013 Southern Ontario Dynamics Day Fields Institute

Angular Momentum Conservation

Thought Experiment I s ω s + I w ω w = 0 .

Horizontally Planar 2-Link Arm y (active) x (passive) I i ω i + I o ω o = 0 .

Horizontally Planar 2-Link Arm: With or Without Damping without damping with damping

Horizontally Planar 2-Link Arm: Global Self-Recovery Self-recovery is global, remembering the winding number.

Mechanical System with an Unactuated Cyclic Variable ▸ Configuration space Q = open subset of R n . q j − V ( q ) with cyclic variable q 1 ▸ Lagrangian L ( q , ˙ q ) = 1 q i ˙ 2 m ij ˙ ∂q 1 = 0 . ∂L ▸ Equations of Motion (EL equations with forces): = − k v ( q 1 ) ˙ d ∂L q 1 dt ∂ ˙ q 1 q a − ∂L ∂q a = u a , a = 2 , ..., n d ∂L dt ∂ ˙ where ▸ − k v ( q 1 ) ˙ q 1 is a viscous damping force ▸ u 2 ,...,u n are control forces

Mechanical System with an Unactuated Cyclic Variable q j − V ( q ) with cyclic variable q 1 ▸ Lagrangian L ( q , ˙ q ) = 1 q i ˙ 2 m ij ˙ ∂q 1 = 0 . ∂L ▸ Equations of Motion (EL equations with forces): = − k v ( q 1 ) ˙ d ∂L q 1 q 1 dt ∂ ˙ q a − ∂L ∂q a = u a , a = 2 , ..., n. d ∂L dt ∂ ˙ ▸ Without damping ( k v = 0 ) q 1 = 0 ⇒ ∂L q 1 = conserved . d ∂L dt ∂ ˙ ∂ ˙

Mechanical System with an Unactuated Cyclic Variable q j − V ( q ) with cyclic variable q 1 ▸ Lagrangian L ( q , ˙ q ) = 1 q i ˙ 2 m ij ˙ such that ∂q 1 = 0 . ∂L ▸ Equations of Motion (EL equations with forces): = − k v ( q 1 ) ˙ d ∂L q 1 q 1 dt ∂ ˙ q a − ∂L ∂q a = u a , a = 2 , ..., n d ∂L dt ∂ ˙ ▸ New conserved quantity with damping q 1 = 0 q 1 dt ( ∂L q 1 + ∫ k v ( x ) dx ) = d q 1 + k v ( q 1 ) ˙ d ∂L ∂ ˙ dt ∂ ˙ 0 q 1 ⇒ ∂L q 1 + ∫ k v ( x ) dx = conserved ∂ ˙ �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� 0 damping-added momentum ▸ ∫ k v ( x ) dx = ∫ 0 k v ( x ) ˙ xdt = (−) impulse due to friction. q 1 ( t ) t 0

Damping-Induced Self-Recovery Phenomenon Theorem (Chang and Jeon [2013, ASME J. DSMC]) Let µ = ∂L q 1 + ∫ q 1 k v ( x ) dx = m 1 i ( q ( t )) ˙ q i ( t ) + ∫ q 1 ( t ) k v ( x ) dx. ∂ ˙ 0 0 Let f ( q 1 ) = ∫ k v ( x ) dx − µ such that q 1 0 Suppose controls u a ( t ) ’s ( a = 2 ,...,n ) are chosen such that q a ( t ) ’s ( a = 2 ,...,n ) are bounded and lim q a ( t ) = 0 for all t →∞ ˙ a = 2 ,...,n . Then, 1. lim t →∞ q 1 ( t ) = q 1 e . q i ( 0 ) = 0 for all 2. If the initial condition is such that ˙ i = 1 ,...,n , then lim t →∞ q 1 ( t ) = q 1 ( 0 ) .

Sketch of Proof for Constant k v with µ = 0 Equation for q 1 q 1 0 = m 1 i ˙ q i ( t ) + ∫ k v dx = m 1 i ˙ q i ( t ) + k v q 1 0 q 1 = − k v q 1 + (− 1 ⇒ ˙ n q a ) , ∑ m 1 a ˙ m 11 m 11 a = 2 q i ( 0 ) = 0 for all i = 1 ,...,n and q 1 ( 0 ) = 0 . Hence, where ˙ q a = 0 ∀ a = 2 ,...,n ⇒ lim t → ∞ q 1 ( t ) = 0 = q 1 ( 0 ) . t → ∞ ˙ lim Remark: Damping coefficient k v ( q 1 ) does not have to be a non-negative function. For example, k v ( q 1 ) = 1 + 4cos ( q 1 ) shows self-recovery for µ = 0 .

Damping-Induced Bound y (active) x (passive) Suppose q 1 k v ( x ) dx = ∞ , q 1 k v ( x ) dx = −∞ . q 1 → ∞ ∫ q 1 → −∞ ∫ lim lim 0 0 If controls u a ( t ) ’s are chosen such that m 11 ( q ( t )) is bounded above and below by two positive numbers and m 1 a ( q ( t )) ’s and q a ( t ) ’s are bounded where a = 2 ,...,n , then q 1 ( t ) is also bounded. ˙

Damping-Induced Bound for Horizontally Planar 2-Link Arm θ 2 is bounded. The motion of Link 1 ( θ 1 ) is bounded when ˙

Real Experiment

Several Unactuated Cyclic Variables Link 2 ( θ 2 ) is actuated and Links 1 and 3 ( θ 1 ,θ 3 ) are unactuated but under friction.

Self-Recovery Seems to Occur Only for Linear Friction Cubic friction F = − kv 3 .

Summary ▸ Viscous damping force breaks symmetry, so the corresponding momentum is no longer conserved. ▸ Exists a new conserved quantity called damping-added momentum . ▸ Damping-induced self-recovery is global. ▸ Damping puts a bound on range of the unactuated variable. ▸ References: ▸ D.E. Chang and S. Jeon, “Damping-induced self recovery phenomenon in mechanical systems with an unactuated cyclic variable,” ASME Journal of Dynamic Systems, Measurement, and Control, 135(2), 2013. http://dx.doi.org/10.1115/1.4007556 ▸ D.E. Chang and S. Jeon, “On the damping-induced self-recovery phenomenon in mechanical systems with several unactuated cyclic variables,” J. Nonlinear Science, Submitted. http://arxiv.org/abs/1302.2109