The power dissipation method and kinematic reducibility of multiple-model robotic systems

Abstract

Abstract-This paper develops a formal connection between the power dissipation method (PDM) and Lagrangian mechanics, with specific application to robotic systems. Such a connection is necessary for understanding how some of the successes in motion planning and stabilization for smooth kinematic robotic systems can be extended to systems with frictional interactions and overconstrained systems. We establish this connection using the idea of a multiple-model system, and then show that multiple-model systems arise naturally in a number of instances, including those arising in cases traditionally addressed using the PDM. We then give necessary and sufficient conditions for a dynamic multiple-model system to be reducible to a kinematic multiple-model system. We use this result to show that solutions to the PDM are actually kinematic reductions of solutions to the Euler-Lagrange equations. We are particularly motivated by mechanical systems undergoing multiple intermittent frictional contacts, such as distributed manipulators, overconstrained wheeled vehicles, and objects that are manipulated by grasping or pushing. Examples illustrate how these results can provide insight into the analysis and control of physical systems.

Feedback Control Methods for Distributed Manipulation Systems that Involve Mechanical Contacts

“…Since the method is a quasi-static modeling method, it produces first-order governing equations, instead of second-order equations that are associated with the Lagrange equations. However, in other work (see Murphey and Burdick 2003) we have shown that these equations of motion are nonsmooth "kinematic reductions" of the complete Lagrangian dynamical models. That is, there is a formal relationship between the equations generated by the PDM analysis and those generated by Lagrangian analysis.…”

“…These experimental observations led us to explicitly incorporate frictional and discontinuous contact effects into the analysis and control of distributed manipulation-and the related case of overconstrained wheeled vehicles (Murphey and Burdick 2000) and overconstrained mechanical systems in general (Murphey and Burdick 2003). We show that under very simple and general assumptions on the friction model, the PFF approach generically leads to unstable systems when implemented on actual distributed manipulation arrays that have frictional contact (see Section 3.2).…”

Feedback Control Methods for Distributed Manipulation Systems that Involve Mechanical Contacts

“…Since the method is a quasi-static modeling method, it produces first-order governing equations, instead of second-order equations that are associated with the Lagrange equations. However, in other work (see Murphey and Burdick 2003) we have shown that these equations of motion are nonsmooth "kinematic reductions" of the complete Lagrangian dynamical models. That is, there is a formal relationship between the equations generated by the PDM analysis and those generated by Lagrangian analysis.…”

“…These experimental observations led us to explicitly incorporate frictional and discontinuous contact effects into the analysis and control of distributed manipulation-and the related case of overconstrained wheeled vehicles (Murphey and Burdick 2000) and overconstrained mechanical systems in general (Murphey and Burdick 2003). We show that under very simple and general assumptions on the friction model, the PFF approach generically leads to unstable systems when implemented on actual distributed manipulation arrays that have frictional contact (see Section 3.2).…”

Modeling and control of multiple-contact manipulation without modeling friction

“…However, we point out here that if one is using a discrete time controller (with one's favorite continuation algorithm, such as zero-order holds) the control is open loop in between controller updates. It was already shown in [16] that U, U -reductions are not affected by occasional discontinuities. By the exact same logic, the systems considered here are reducible in discrete time closedloop if they satisfy the requirements to be reducible under the tests of Lemma 4.…”

Kinematic reductions for uncertain mechanical contact

“…It was already shown in ref. [18] for two control families U and U that U, U -reductions are not affected by separable discontinuities. By the exact same logic, the systems considered here are reducible in discrete time closed-loop if they satisfy the requirements to be reducible under the tests of Theorem 4.2.…”

Feedback Control Methods for Distributed Manipulation Systems that Involve Mechanical Contacts

“…Since the method is a quasi-static modeling method, it produces first-order governing equations, instead of second-order equations that are associated with the Lagrange equations. However, in other work (see Murphey and Burdick 2003) we have shown that these equations of motion are nonsmooth "kinematic reductions" of the complete Lagrangian dynamical models. That is, there is a formal relationship between the equations generated by the PDM analysis and those generated by Lagrangian analysis.…”

“…These experimental observations led us to explicitly incorporate frictional and discontinuous contact effects into the analysis and control of distributed manipulation-and the related case of overconstrained wheeled vehicles (Murphey and Burdick 2000) and overconstrained mechanical systems in general (Murphey and Burdick 2003). We show that under very simple and general assumptions on the friction model, the PFF approach generically leads to unstable systems when implemented on actual distributed manipulation arrays that have frictional contact (see Section 3.2).…”

Modeling and control of multiple-contact manipulation without modeling friction

“…However, we point out here that if one is using a discrete time controller (with one's favorite continuation algorithm, such as zero-order holds) the control is open loop in between controller updates. It was already shown in [16] that U, U -reductions are not affected by occasional discontinuities. By the exact same logic, the systems considered here are reducible in discrete time closedloop if they satisfy the requirements to be reducible under the tests of Lemma 4.…”

Kinematic reductions for uncertain mechanical contact

“…It was already shown in ref. [18] for two control families U and U that U, U -reductions are not affected by separable discontinuities. By the exact same logic, the systems considered here are reducible in discrete time closed-loop if they satisfy the requirements to be reducible under the tests of Theorem 4.2.…”

Feedback Control Methods for Distributed Manipulation Systems that Involve Mechanical Contacts

“…Since the method is a quasi-static modeling method, it produces first-order governing equations, instead of second-order equations that are associated with the Lagrange equations. However, in other work (see Murphey and Burdick 2003) we have shown that these equations of motion are nonsmooth "kinematic reductions" of the complete Lagrangian dynamical models. That is, there is a formal relationship between the equations generated by the PDM analysis and those generated by Lagrangian analysis.…”

“…These experimental observations led us to explicitly incorporate frictional and discontinuous contact effects into the analysis and control of distributed manipulation-and the related case of overconstrained wheeled vehicles (Murphey and Burdick 2000) and overconstrained mechanical systems in general (Murphey and Burdick 2003). We show that under very simple and general assumptions on the friction model, the PFF approach generically leads to unstable systems when implemented on actual distributed manipulation arrays that have frictional contact (see Section 3.2).…”

Modeling and control of multiple-contact manipulation without modeling friction

“…However, we point out here that if one is using a discrete time controller (with one's favorite continuation algorithm, such as zero-order holds) the control is open loop in between controller updates. It was already shown in [16] that U, U -reductions are not affected by occasional discontinuities. By the exact same logic, the systems considered here are reducible in discrete time closedloop if they satisfy the requirements to be reducible under the tests of Lemma 4.…”

Kinematic reductions for uncertain mechanical contact

“…It was already shown in ref. [18] for two control families U and U that U, U -reductions are not affected by separable discontinuities. By the exact same logic, the systems considered here are reducible in discrete time closed-loop if they satisfy the requirements to be reducible under the tests of Theorem 4.2.…”