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Research Papers

# Optimal Splines for Rigid Motion Systems: Benchmarking and Extensions

[+] Author and Article Information
B. Demeulenaere

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

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

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

Qiu et al.  call the spline coefficients control points, which should not be confused with spline knots.

This choice might have been inspired by the nondifferentiability of the problem.

That is, the optimization variable $d$ is replaced by a new optimization variable $d̃$ with $N$ independent spline coefficients that represent a quintic spline that automatically complies with the five linear equality constraints 5,5.

Qiu et al.  provide no information about the time increment with which $θ(τ)$ and its derivatives are sampled to evaluate the objective function 5 and the constraints 5,5.

Another detail important for obtaining an exact benchmark is that the upper limits $Cv,mt$ and $Cj,mt$ in Eqs. 5,5 are replaced by $Cv,Qiu$ and $Cj,Qiu$ in Eqs. 9,9.

Note, however, that these values might be judged “large” compared with the 0.39% improvement with respect to the modified trapezoidal law that was reported by Qiu et al.

Feedforward computed from a trajectory that is continuous up to the jerk, as is a quartic spline.

By convention, infinity is returned as the optimal value if the problem is infeasible, that is, if no optimization variable can be found (exists) that satisfies all equality and inequality constraints (12).

$⌈⋅⌉$ denotes the ceil operator, that is, rounding to the nearest larger integer.

In the definition of $δd$, $Tbisec$ is compared with $Td,Lam$ since the discrete solution is the one actually implemented in the control system.

Depending on the problem data, the results of Lambrechts et al.  are improved by up to 6%.

Lambrechts et al.  remark that, in principle, higher than fourth-order trajectories can be planned by means of their algorithm, but this is considered impractical due to the large increase in complexity.

The slope of the regression line in Fig. 8 is minus one.

The slope of the regression line in Fig. 8 is 1.4.

In this particular case, two orders of magnitude bigger.

Since $f(y)=y2$ is a convex function, Eq. 17 is convex in $Mm(ti)$$(Mm(ti)=I⋅r⋅s̈(ti)+c⋅r⋅ṡ(ti)+Mc)$: $E=∑i=1g+1Mm(ti)2$. Since the composition with an affine mapping preserves the convexity of a function, $E$ is convex in $x$, where the affine dependency of $Mm(ti)$ on $x$ follows from the affine dependency of $s̈(ti)$ and $ṡ(ti)$ on $x$ (dimensional counterpart of Eq. 4).

J. Mech. Des 131(10), 101005 (Sep 03, 2009) (13 pages) doi:10.1115/1.3201991 History: Received June 02, 2008; Revised July 01, 2009; Published September 03, 2009

## Abstract

This paper illustrates the power and versatility of the convex programming framework for optimal spline synthesis that was developed in a companion paper (Demeulenaere, Pipeleers, De Caigny, Swevers, De Schutter, and Vandenberghe, 2009, “Optimal Spines for Rigid Motion Systems: A Convex Programming Framework”, ASME J. Mech. Des., 131, p. 101005.). Two case studies concerning rigid motion systems illustrate the ability of the framework to improve upon recent (2005) literature results: (i) a numerical optimization study concerning kinematic optimization of uniform quintic splines for cam systems and (ii) an analytical study concerning time optimal quartic splines for motion systems driven by servomotors and subject to kinematic constraints. In a third study, the versatility of the framework is illustrated by generating time optimal and time-energy optimal motions for a rigid servomotor driven system under torque constraints. Based on these three case studies, the convex programming framework of the companion paper is extended with the following generic aspects: (i) a bisection to generate time optimal motions, (ii) a direct expression of the upper and lower bounds on motor torque, and (iii) a convex quadratic energy objective function for servomotor driven systems.

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## Figures

Figure 1

The quintic spline θ(τ) and its first three derivatives, obtained as a solution of Eq. 5 by Qiu (4)

Figure 2

The trajectories of ping θ(4)(τ) and puff θ(5)(τ) for solutions 1 (Qiu , solid line), 2 (dashed line), and 3 (dash-dotted line) indicated in Fig. 3. The thick vertical line at τ=0.5 indicates the infinite puff value at this location.

Figure 3

Trade-off between the peak value Cp of puff and acceleration Ca for quintic splines with peak velocity Cv≤Cv,Qiu and peak jerk Cj≤Cj,Qiu and complying with the (incomplete) boundary constraints 5,5. Results for N=7 (dashed line) and N=350 (solid line). The design of Qiu (4) is indicated by the cross labeled 1. Table 1 summarizes the properties of the competing designs 1, 2, and 3.

Figure 4

Trajectories of ping θ(4)(τ) and puff θ(5)(τ) for solution 1 indicated in Fig. 3 (Qiu , solid line) and the solutions obtained through regularization of the solutions 2 (dashed line) and 3 (dash-dotted line). The thick vertical line at τ=0.5 indicates the infinite puff value at this location.

Figure 5

Solutions of Eq. 11 for case 1 (Table 2): discrete solution of Lambrechts (dashed line) and our bisection solution (solid line, end point marked by a circle)

Figure 6

Solutions of Eq. 11 for case 2 (Table 2): discrete solution of Lambrechts (dashed line) and our bisection solution (solid line, end point marked by a circle)

Figure 7

Analytical solution of Eq. 15: s(t) and its first two derivatives, as well as the corresponding motor torque Mm(t). The horizontal dashed lines in (b) and (d) indicate the upper and lower limits expressed by Eqs. 15,15, respectively.

Figure 8

(a) Relative error εt(g) between the analytical solution Ta∗ of Eq. 15 and the solution Tb∗ determined by bisection using g internal knots. (b) Computational time required for the whole bisection process as a function of g. The solid lines mark linear regression lines.

Figure 9

Time-energy trade-off. Emin(T)/Emin(Ta∗) (solid line) and Mrms(T)/Mrms(Ta∗) (dashed line) as a function of T/T∗. Cases 1, 2, and 3 defined in Sec. 4 and presented in Table 5 are indicated.

Figure 10

s(t) and its first two derivatives, as well as the corresponding motor torque Mm(t) of time optimal (case 1, solid line) and time-energy optimal trajectories (case 2, dash-dotted line; case 3, dashed line) corresponding to the numerical data of Table 4. Table 5 summarizes the main properties. The end point of the trajectories is marked by a circle (case 1), a diamond (case 2), or an asterisk (case 3). The horizontal dashed lines in (b) and (d) indicate the upper and lower limits expressed by Eqs. 16,16, respectively.

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