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Projective Constraint Stabilization for a Power Series forward dynamics Solver

JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME 2013年第3期135卷 031004-031004-9页

作者： Milenkovic, Paul Univ Wisconsin Dept Elect & Comp Engn Madison WI 53706 USA

A power series expression for the forward dynamics of a closed kinematic chain provides an explicit time-step update of the system state. The resulting numerical differential equation solver applies kinematic constraints to the power series terms for acceleration and higher derivatives of motion. Integrating acceleration determines velocity and position time histories that approximate the constraints to a high degree of precision when using a high order of the expansion. When high precision is not required, a lower order achieves shorter computation times, but that condition results in violation of the constraints in the absence of any correction. Projecting the velocities and positions onto the constraint manifold after each time step produces step changes. This paper determines which choices of linear subspace for this projection give step changes that are equal to the residues of truncating the power series solution for the kinematic portion of the problem. The limit of that power series gives position and velocity time histories that approximate the dynamics while giving an exact kinematic solution. Thus projection onto the constraints in this procedure determines sample values of an underlying solution for the motion trajectories, where that underlying solution is continuous in both velocity and position and also satisfies the kinematic constraints at all times. This property is confirmed by numerical simulation of a Clemens constant-velocity coupling.

关键词： kinematic differential equation closed kinematic chain power series Taylor series screw theory forward-dynamics problem differential-algebraic equation (DAE) constraint stabilization projection explicit time step

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Multi-Integral Method for Solving the forward dynamics of Stiff Multibody Systems

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JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME 2013年第5期135卷 051014-051014页

作者： Milenkovic, Paul Univ Wisconsin Dept Elect & Comp Engn Madison WI 53706 USA

The Hermite-Obreshkov-Pade (HOP) procedure is an implicit method for the numerical solution of a system of ordinary differential equations (ODEs) applicable to stiff dynamical systems. This procedure applies an Obreshkov condition to multiple derivatives of the system state vector, both at the start and end of a time step in the numerical solution. That condition is shown to be satisfied by the Hermite interpolating polynomial that matches the state vector and its derivatives, also at the start and end of a time step. The Hermite polynomial, in turn, can be specified in terms of the system state and its derivatives at the start of a step together with a collection of free parameters. Adjusting these free parameters to minimize magnitudes of the ODE residual and its derivatives at the end of a step serves as a proxy for matching the system state and its derivatives. A high-order Taylor expansion at the start of a time step interval models the residual and its derivatives over the entire interval. A variant of this procedure adjusts those parameters to match integrals of the system state over the duration of that interval. This is done by minimizing magnitudes of integrals of the ODE residual calculated from the extrapolating Taylor-series expansion, a process that avoids the need to determine integration constants for multiple integrals of the state. This alternative method eliminates the calculation of high-order derivatives of the system state and hence avoids loss in accuracy from floating-point round off. Numerical performance is evaluated on a dynamically unbalanced constant-velocity (CV) coupling having a high spring rate constraining shaft deflection.

关键词： forward-dynamics problem differential-algebraic equation (DAE) numerical integration Hermite-Obreshkov-Pade (HOP) method stiff system implicit explicit single-step method algorithmic differentiation constant-velocity coupling

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Numerical Solution of Stiff Multibody Dynamic Systems Based on Kinematic Derivatives

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JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME 2014年第6期136卷 061001-061001页

作者： Milenkovic, Paul Univ Wisconsin Dept Elect & Comp Engn Madison WI 53706 USA

The Hermite-Obreshkov-Pade (HOP) method of numerical integration is applicable to stiff systems of differential equations, where the linearization has large range of eigenvalues. A practical implementation of HOP requires the ability to determine high-order time derivatives of the system variables. In the case of a constrained multibody dynamical system, the power series solution for the kinematic differential equation is the foundation for an algorithmic differentiation (AD) procedure determining those derivatives. The AD procedure is extended in this paper to determine rates of change in the time derivatives with respect to variation in the position and velocity state variables of the multibody system. The coefficients of this variation form the Jacobian matrix required for Newton-Raphson iteration. That procedure solves the implicit relations for the state variables at the end of each integration time step. The resulting numerical method is applied to the rotation of a dynamically unbalanced constant-velocity (CV) shaft coupling, where the deflection angle of the output shaft is constrained to low levels by springs of high rate and damping.

关键词： closed kinematic chain screw theory forward-dynamics problem differential-algebraic equation (DAE) numerical integration Hermite-Obreshkov-Pade (HOP) method stiff system implicit method single-step method automatic differentiation algorithmic differentiation constant-velocity coupling

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