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Research Papers

Motion Planning of Uncertain Ordinary Differential Equation Systems

[+] Author and Article Information
Joe Hays

Control Systems Branch,
Spacecraft Engineering Division,
Naval Center for Space Technology,
US Naval Research Laboratory,
Washington, DC 20375
e-mail: joehays@vt.edu

Adrian Sandu

Computational Science Laboratory,
Computer Science Department,
Virginia Tech,
Blacksburg, VA 24061
e-mail: sandu@cs.vt.edu

Corina Sandu

Advanced Vehicle Dynamics Laboratory,
Mechanical Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: csandu@vt.edu

Dennis Hong

Robotics and Mechanisms Laboratory,
Mechanical Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: dhong@vt.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 14, 2012; final manuscript received March 1, 2014; published online May 13, 2014. Assoc. Editor: Jozsef Kovecses.

J. Comput. Nonlinear Dynam 9(3), 031021 (May 13, 2014) (15 pages) Paper No: CND-12-1049; doi: 10.1115/1.4026994 History: Received March 14, 2012; Revised March 01, 2014

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and underactuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it is not accounted for in a given design. In this work uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and underactuated systems, prescribes deterministic actuator inputs that yield uncertain state trajectories. The inverse dynamics formulation is the dual to that of forward dynamics, and is only applicable to fully actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to underactuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories that yield uncertain unactuated states and uncertain actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space.

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Figures

Grahic Jump Location
Fig. 1

A simple illustration of an uncertain fully actuated motion planning problem. The forward dynamics based formulation aims to determine an effort optimal motion plan and the inverse dynamics based formulation aims to determine a time optimal motion plan. Both problems are subject to input wrench and geometric collision constraints. This system is an uncertain system due to the uncertain mass of the payload.

Grahic Jump Location
Fig. 2

The effort optimal configuration time histories for the deterministic serial manipulator pick-and-place problem. This optimal solution resulted in a J = 2770 (Nm)2 design. (The initial configuration starts at the target ‘x’ on the left and finishes at the ‘x’ target on the right while progressively darkening from light gray to black.)

Grahic Jump Location
Fig. 3

The effort optimal uncertain end-effector Cartesian position time history for the uncertain serial manipulator pick-and-place problem based on the uncertain forward dynamics NLP. The mean and bounding time histories μyi± αi·σyi are displayed ∀i = {1,2} with αi = 1. This optimal solution resulted in a J = 3530 (Nm)2 design.

Grahic Jump Location
Fig. 4

The terminal variance optimal uncertain end-effector Cartesian position time history for the uncertain serial manipulator pick-and-place problem based on the uncertain forward dynamics NLP. The mean and bounding time histories μyi± αi·σyi are displayed ∀i = {1,2} with αi = 1. This optimal solution resulted in a J = 5910 (Nm)2 design.

Grahic Jump Location
Fig. 5

The time optimal input wrench time histories for the deterministic serial manipulator pick-and-place problem based on the uncertain inverse dynamics NLP. This optimal solution resulted in a tf = 1.12 s.

Grahic Jump Location
Fig. 6

The time optimal uncertain input wrench time histories for the uncertain serial manipulator pick-and-place problem based on the uncertain inverse dynamics NLP. Each input wrench is displaying its mean value and bounding time histories μτi± αi·στi with αi = 1,∀i = {1,2}. This optimal solution resulted in a tf = 1.2 s.

Grahic Jump Location
Fig. 7

The final optimal configuration time history of the uncertain serial manipulator pick-and-place application involving collision avoidance and actuator constraints design with the uncertain inverse dynamics NLP. (The initial configuration starts at the target ‘x’ on the left and finishes at the ‘x’ target on the right while progressively darkening from light gray to black.)

Grahic Jump Location
Fig. 8

A simple illustration of the underactuated uncertain hybrid dynamics motion planning formulation. This problem aims to determine a power optimal motion plan to lift the pendulum from the initial hanging configuration to an inverted vertical configuration when subject to input wrench and terminal condition constraints. This is an uncertain system due to the uncertain mass of the payload.

Grahic Jump Location
Fig. 9

The power optimal configuration time history for the deterministic inverting double pendulum. This optimal solution resulted in a 1060 W design. (The initial configuration starts with the double pendulum in the down position and swings up to the vertical while progressively darkening from light gray to black.)

Grahic Jump Location
Fig. 10

The uncertain input wrench time history for the deterministically designed motion plan applied to an uncertain inverting double pendulum (where μτ± αστ with α = 1). The presence of the uncertainty results in both the maximum and minimum input limits being exceeded.

Grahic Jump Location
Fig. 11

The joint time histories for the deterministically design motion plan applied to an uncertain inverting double pendulum (where μq2±ασq2 with α = 1). The presence of the uncertainty results in the expected terminal error condition not being satisfied with excessive variance.

Grahic Jump Location
Fig. 12

The power optimal configuration time history for the uncertain inverting double pendulum based on uncertain hybrid dynamics NLP, where {μyi - αi·σyi(solid),μyi + αi·σyi(dash - dot)},∀i = {1,2} with αi=1. This optimal solution resulted in a 310 W design. (The initial configuration starts with the double pendulum in the down position and swings up to the vertical while progressively darkening from light gray to black.)

Grahic Jump Location
Fig. 13

The uncertain input wrench time history resulting from the motion plan generated by the new uncertain hybrid dynamics NLP (where μτ±αστ with α = 1). Both the maximum and minimum input limits were satisfied, in a weighted standard deviation sense, for all systems within the probability space.

Grahic Jump Location
Fig. 14

The joint time histories resulting from the motion plan generated by the new uncertain hybrid dynamics NLP (where μq2± ασq2 with α = 1). The resulting terminal error variance satisfies the specification σe(tf)2 = 0.0032≤σ¯e(tf)2 = 0.01m2.

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