Cubic Spline Interpolation for Generating Smooth End-Effector Trajectories through Waypoints

Project info

• Field of studyRobotics and Mechatronics
• LevelUndergraduate thesis (S1)
• Method typeCommunications and Forecasting
• Estimated build time± 3 working days
• Toolsspline, pchip, interp1; csaps (Curve Fitting Toolbox)
• PriceUS$19 - US$119

Research goal

Analyse the end-effector position using spline interpolation and examine its sensitivity to the number of waypoints and the motion segment duration.

Variables

Example result

• 11Knots
• 0.282Max error linear
• 0.226Max error spline
• 0.263Max error PCHIP
• 0.083RMSE spline
• 0.662Max slope error spline

With 11 knots the cubic spline cuts the maximum error of End-effector position from 0.282 (linear) to 0.226 with an RMSE of 0.0833; PCHIP preserves shape at an error of 0.263 and the spline keeps a smooth first derivative.

Computed from illustrative synthetic data, not from a real study.

About Spline Interpolation

Spline interpolation joins data points with smooth piecewise cubic polynomials, while a smoothing spline (csaps) filters noise. It is used to fill between measurement points, compute derivatives, or build smooth curves from experimental data.

Steps

1. 1Prepare sorted x points and y values
2. 2Interpolate with spline, pchip, and linear
3. 3Build a dense evaluation grid
4. 4Compare error against the true function if known
5. 5Use csaps for noisy data with a smoothing parameter
6. 6Plot the resulting curves and data

Assumptions to check

• The x points are unique and sorted
• An interpolating spline follows data exactly so it is unsuitable for noisy data
• The underlying function is reasonably smooth

Reading and reporting the results

Report the interpolation error (RMSE) for each method and the reason for the spline type chosen. For noisy data, explain how the smoothing parameter was chosen.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

rng(9);
truef = @(x) sin(x) + 0.3*x;
number_of_waypoints = linspace(0, 10, 12)';                      % coarse sample points
end_effector_position = truef(number_of_waypoints);
xi = linspace(0, 10, 500)';

ys = spline(number_of_waypoints, end_effector_position, xi);
yp = pchip(number_of_waypoints, end_effector_position, xi);
yl = interp1(number_of_waypoints, end_effector_position, xi, 'linear');
rm = @(a) sqrt(mean((a - truef(xi)).^2));
fprintf('RMSE spline = %.4f, pchip = %.4f, linear = %.4f\n', rm(ys), rm(yp), rm(yl));

xn = linspace(0, 10, 80)';
yn = truef(xn) + 0.2*randn(size(xn));               % noisy data
sp = csaps(xn, yn, 0.9);
ysm = fnval(sp, xi);
fprintf('RMSE smoothing spline = %.4f\n', rm(ysm));

figure;
subplot(1,2,1);
plot(xi, truef(xi), 'k', xi, ys, 'r', xi, yp, 'b--', xi, yl, 'g:', number_of_waypoints, end_effector_position, 'ko');
legend('True', 'Spline', 'PCHIP', 'Linear', 'Data'); title('Interpolation');
subplot(1,2,2);
plot(xn, yn, '.', xi, ysm, 'r', 'LineWidth', 1.5); title('Smoothing spline');

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions