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Research Papers

# Optimal Splines for Rigid Motion Systems: A Convex Programming Framework

[+] Author and Article Information
B. Demeulenaere

Atlas Copco Airpower NV, Airtec Division, Boomsesteenweg 957, B-2610 Wilrijk, Belgium

G. Pipeleers, J. De Caigny, J. Swevers, J. De Schutter

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300B, B-3001 Leuven, Belgium

L. Vandenberghe

Department of Electrical Engineering, University of California Los Angeles, 68-119 Engineering IV, Los Angeles, CA 90095

Spline knots, coefficients, and degrees are rigorously defined in Sec. 1.

If the knot sequence is increasing but not strictly increasing (that is, coincident knots are present), the continuity condition 2 has to be relaxed.

Sparse problems have constraint functions that depend on a few variables only.

A formal definition of convexity is provided in Appendix .

Also nonuniform distributions, as in Ref. 4, can be considered.

Figure 1 illustrates this for a cubic $k=3$ spline: Its $(k−1)st$ (second) order derivative is piecewise-linear.

Provided that the driven mechanism is sufficiently stiff (although a motion system may be assumed to be rigid in the design phase, it is never infinitely rigid in practice).

As can be seen by counting the number of jumps in $θ(m+1)(τ)$ for each of the considered $m$.

Observe that the factor $1/(g+1)$ in Eq. 8 is a rational number for integer $g$.

The values of $τ̂mj$ are numerically found by detecting jumps of $θ(m+1)(τ)$ in the solution of Eq. 23.

The floor operator $⌊x⌋$ rounds $x$ to the nearest integer, less than or equal to $x$.

Being based on the linear interpolation 27, Eq. 28 is an approximation of the exact constraint 25. However, given that $δτ$ is typically very small, the approximation is sufficiently accurate. If judged necessary, Eq. 25 can be expressed exactly as a linear, yet more tedious to derive, equality constraint in $x$.

This conjecture does not contradict the convexity of the problem: Convexity guarantees that the globally optimal objective value $f0∗$ will be found, but there may be many vectors $x∗$ for which $f0(x∗)=f0∗$. Which of these $x∗$ is actually obtained depends on the particular algorithm used.

While in the form 31, the one-norm constitutes a nonlinear, nondifferentiable function due to the presence of the absolute value. It can easily be transformed into a linear objective function through the introduction of auxiliary variables and additional linear inequality constraints, similar to the transformation discussed in Sec. 3. More details are provided in, e.g., Ref. 17.

Convex quadratic programs are discussed more thoroughly in the companion paper (15).

Roughness is defined as the mean square of the $(k−1)/2nd$-order derivative, and hence, corresponds to a convex quadratic objective function of the general type 33.

J. Mech. Des 131(10), 101004 (Sep 03, 2009) (11 pages) doi:10.1115/1.3201977 History: Received June 02, 2008; Revised June 24, 2009; Published September 03, 2009

## Abstract

This paper develops a general framework to synthesize optimal polynomial splines for rigid motion systems driven by cams or servomotors. This framework is based on numerical optimization, and has three main characteristics: (i) Spline knot locations are optimized through an indirect approach, based on providing a large number of fixed, uniformly distributed candidate knots; (ii) in order to efficiently solve the corresponding large-scale optimization problem to global optimality, only design objectives and constraints are allowed that result in convex programs; and (iii) one-norm regularization is used as an effective tool for selecting the better (that is, having fewer active knots) solution if many equally optimal solutions exist. The framework is developed and validated based on a double-dwell benchmark problem for which an analytical solution exists.

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## Figures

Figure 1

An example of a cubic spline with knot sequence {0.0,0.64,0.65,1.00,1.91,2.68,3.25,4.21,5.00,5.72,6.28}. Active knots are indicated with a dashed vertical line, while dotted vertical lines denote the (redundant) inactive knots τ=1.00 and τ=5.00.

Figure 2

Piecewise-linear parametrization of θ(k−1)(τ), where δτ=2π/(g+1)

Figure 3

Analytical benchmark problem solutions for m=0,1,2,3. The solution is a polynomial spline of degree m+1 with g=m internal knots.

Figure 4

Relative error εp (%) on the obtained peak absolute value of θ(4)(τ) for m=3, and 50 logarithmically spaced even g-values between 10 and 1000

Figure 5

Numerically computed knots τ̂m2 and τ̂m3 (x) for m=3, and 50 logarithmically spaced even or odd g-values between 10 and 1001: (a) τ̂m2 for even values of g, (b) τ̂m3 for even values of g, (c) τ̂m2 for odd values of g, and (d) τ̂m3 for odd values of g. The analytical values τm2∗ and τm3∗ are indicated by the dashed-dotted line, while the solid lines indicate the error bounds derived from Eq. 24.

Figure 6

Analytical benchmark problem revisited (g=1000, m=2): nonmonotonous solution (dashed line), monotonous solution without one-norm regularization (solid line), and monotonous solution with one-norm regularization (dashed-dotted line). For τ≤2.64, the latter two solutions coincide. The circles in (a) indicate the precision points 26.

Figure 7

Analytical benchmark problem revisited (g=1000, m=2): (a) jerk derivative on logarithmic scale for monotonous solution without one-norm regularization, (b) monotonous solution with one-norm regularization, and monotonous solutions with relaxed one-norm regularization: (c) ε=1e−12, (d) ε=1e−8, (e) ε=1e−4, and (f) ε=1e−2

Figure 8

The hat functions β0(τ), βi(τ)(1≤i≤g), and βg+1(τ)

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