On Coordinate Choices, Regularization, and Degree of

Nonlinearity for Dynamical Systems

John L. Junkins1 , Puneet Singla1, John E. Hurtado[1], and Andrew J. Sinclair[2]

Summary

The nonlinearity of dynamical systems depends on the physics of the system as well as the decisions made by the analyst in choosing coordinates to represent the system. An index for “measuring nonlinearity” is discussed and some classical and novel choices of coordinates are considered for example dynamical systems. We consider cases wherein the same motion is represented using more than one coordinate choice; it is shown that our nonlinearity index is useful for coordinate selectionif one is interested in obtaining near-linear forms. Whereas an infinity of coordinate choices is feasible for most problems, we review problems for which some coordinate choices result in non-linear, singular differential equations and other choices of coordinates lead to regularized dynamics over much larger domains of the spate space. Of particular significance, we show how advantages analogous to the case of classical rigid body dynamics for general Lagrangian dynamics can be realized by choosing to describe angular velocity of a n-dimensional abstract rigid body by orthogonal components along the moving body fixed axes. We make use of the Cayley Transform to introduce a quasi-coordinate description of velocity along the rotating axes of an “n-dimensional virtual rigid body” for general motions, and demonstrate the advantages of this transformation vis a vis improving the linearity of the resulting dynamical system representation.

Introduction

Recent literature has considered measures of the nonlinearity of departure motion from a given reference trajectory, as a function of coordinate selections. A significant effort over the history of analytical mechanics has been a quest to find judicious coordinates in particular problems, or to establish broadly applicable coordinate transformation methodology for classes of problems. This quest led Hamilton and Jacobi to develop canonical transformation theory, Euler and Hamilton to discover the quaternion, Lagrange and Poisson to invent the method of variation of parameters, Cayley and Rodrigues to introduce novel representations of rotations [1], and recent studies, e.g., Sinclair, Hurtado, Singla, and Junkins [2-6] to formulate regularizing coordinate transformations with important consequences in dynamical systems broadly. Coordinate transformations generally affect the nonlinearity of a dynamical system, but there is no universal consensus on what we mean when we say “System A is more nonlinear than System B.” The next portion of this paper will briefly discuss this issue: How does one measure nonlinearity?

Nonlinearity Measure for Dynamical Systems

Consider the nonlinear initial value problem:

(1)

is the solution of Eq. (1), with nominal initial state; is a differentiable vector-valued function and . We assume that the state vector has been non-dimensionalized so all of the elements of are of comparable numerical size. Consider the varied set of worst-case initial conditions, approximately uniformly distributed on a prescribed n-dimensional sphere or other closed surface:

(2)

where . Note the exact nonlinear departure motion trajectory can be obtained by solving for for each of the varied initial conditions as the solution of Eq. (1) with the nominal initial condition replaced by . The linear prediction of the nonlinear varied trajectories for all N variations is determined by the initial variations from the nominal state transition matrix as

(3)

A key issue, vis a vis nonlinearity, is the accuracy with which the nonlinear variations are predicted by the linear approximation of Eq. (3). In Ref. [1], the following nonlinearity index was introduced:

(4)

is the largest variation of , over the N worst-case set of initial conditions. Qualitatively < 0.01 corresponds to a near-linear behavior, and > 1 corresponds to highly non-linear behavior. Of interest is the evaluation of a coordinate transformation effect on the nonlinearity of the same physical motions: Specifically, we investigate the extent to which several systems are more or less nonlinear depending upon the choice of coordinates.

Rotational Dynamics of Rigid Bodies

Consider the motion of a rigid body containing an imbedded orthogonal triad of unit vectors relative to an inertial triad . The orientation of body basis vectors relative to inertial basis vectors is specified by the direction cosine matrix according to; is a proper orthogonal matrix and can be parameterized as a function of three or more coordinates. The time evolution of can be obtained by solving the differential equation linear in angular velocity.

(5)

For minimal representations of , the 3 choices of Table 1 are important examples; including the singularities. The volume of state space where singularities are encountered is decreased by using the Rodrigues or Modified Rodrigues Parameters (MRPs) as compared to Euler Angles. To study the nonlinearity of the representations in Table 1, we prescribe a specific motion defined in terms of the 3-2-1 Euler Angles.

(6)

Referring to Fig 2a, for these large motions we can expect significant departures from the linear approximations. The worst-case surface is prescribed as a sphere of radius () centered at nominal initial motion. The nominal motion chosen passes near the singularity at and as a consequence, nonlinear behavior can be anticipated for the Euler Angle representation. The number N of extreme initial conditions was chosen to ensure that the nonlinearity index converged, in this case, we found N = 200 was more than sufficient. Figs. 2c & 2d shows the variation of maximum non-linear index with sample size N, as anticipated, the nonlinearity index is O(> 1) for the Euler angles and much smaller for other attitude parameters. Fig. 2b shows the time variation of the nonlinearity index for three sets of coordinates. From (Fig. 2), the classical and (especially the) modified Rodrigues parameters are much more nearly linear than are the 321 Euler angles.

In order to make practical use of this idea, the worst case surface represent the reality of that particular application. This idea, has applications for many other situations. For example, in many cases the nonlinearity index grows with time or reveals local nonlinearities that are useful for intuitive insight.

N-Dimensional Rigid Body Motion Representation of General Lagrangian Dynamics

In a recent dissertation [6], we began with the Cayley transform parameterization of a general proper orthogonal matrix

(7)

The vector has elements that lie between , and can be easily shown to reduce for the case to the classical Rodrigues parameters. Thus Eq (7) can be used to associate a rigid body orientation with an arbitrary vector . We denote as the

Table 1 Kinematic Differential Equations for Three Attitude Coordinates

Coordinates / Kinematic Differential Eqs / Range / SingularAt
3-2-1 Euler
Angles / / /
Classical
Rodrigues
Parameters q / / /
Modified
Rodrigues
Parameters  / / /

Figure 2 Nonlinearity Index for Three Attitude Coordinates

Extended Rodrigues Parameters (ERPs). In Ref [2], we show that the ERPs satisfy the generalized kinematic differential equation

(8)

Since Eqs (8) & (9) hold for arbitrary elements of the vector, Eq (8) can be used as a transformation, for a general Lagrangian dynamical system. Note that Eq (8)associates with a general motion a rotational motion of a set of n orthogonal axes with angular velocity . The components of the  vector are therefore an orthogonal quasi-velocity description for a general motion. The  vector can be projected directly onto . The coefficients of linear combination and a version of Lagrange’s eqs are derived for this transformation of a general Lagrangian system into the motion of an n-dimensional rigid body, in Refs. [3-6]. In the Cayley form, Eq (8) is used as definition for the Cayley quasi-velocities, and their relationship to the generalized coordinates and velocities,. Of course, traditionally these variables are represented in vector forms: and . The mapping between the skew-symmetric matrix and vector form has been carried out in a general way [2,3] and is not repeated here.

Consider the dynamics of an elastic pendulum, see Fig. 3. In spherical coordinates , the equations of motion are:

(9)

The geometric and kinematic transformations that define the state variables consistent with the geometry and Eqs (9), and the resulting differential equations of motion are:

(10)

Initial conditions were adopted for the nominal motion, along with m = k = g = 1 and 500 uniformly spaced initial conditions variations were generated on a 6 dimensional initial variation sphere satisfying the worst-case surface constraint: . The 500 consistent variations were used to compute the nonlinearity index. See Fig. 6. As is evident, the set is vastly superior to the set as regards nonlinearity in the vicinity of the chosen nominal trajectory.

Discussion

As is evident, the quasi-coordinate transformation is quite constructive in this examples shown above to characterize the nonlinearity of motion, as described in two or more coordinate choices. Some classical advantages enjoyed in rigid body dynamics can be extended to general Lagrangian systems through the use of the Cayley form. Other examples and the details of the examples are given in [1-4].

Figure 3 Elastic Pendulum Figure 4 Elastic Pendulum Motion

Figure 5 Elastic Pendulum Motion in “Rigid Body Quasi-Coordinates”

Figure 6 Nonlinearity Index as a Function of Time: Two Coordinate Choices

References
  1. Junkins, J. L. and Singla, P. (2004): “How Nonlinear Is It? A Tutorial on Nonlinearity of Orbit and Attitude Dyanmics,” keynote paper at the AAS John L. Junkins Astrodynamics Symp, May 2003, J. Astron Sci, Vol 52, Nos 1 and 2, pp 7-60.
  2. Junkins, J., Kim, Y. (1993): Dynamics and Control Flexible Structures, AIAA Education Series, AIAA, Washington, D. C.
  3. Hurtado J. E. and Sinclair A.J. (2004): “Hamel coefficients for the rotational motion of an N-dimensional rigid body,” Proc. of the Royal Society of London SeriesA, Vol. 460, No. 2052, pp. 3613–3630.
  4. Sinclair A.J. and Hurtado J.E. (2005): “Cayley kinematics and Cayley form of dynamics,” Proc. Royal Soc of London Series A, Vol. 461, No. 2055, pp. 761–781.
  5. Sinclair A.J. and Hurtado J. E. (2004): “Application of the Cayley Form to General Spacecraft Motion,” AAS/AIAA Spaceflight Mechanics Meeting, Hawaii.
  6. Sinclair, A. J. (2005): “Generalization of Rotational Mechanics and Application to Aerospace Systems,” Ph.D. Dissertation, Dept Aero Engr, Texas A&M Univ.

[1]TexasA&MUniversity, Department of Aerospace Engineering, College Station, Texas. 977843-3141, USA

[2]AuburnUniversity, Aerospace Engineering Department, Auburn, Alabama. 36849-5338, USA