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Cubic Spline Interpolation for Smooth Trajectory Generation in Autonomous Motion Systems
Aarush Mandayam, Sriram Aditya Gundla, Elliott Tammaro
TL;DR
Piecewise-linear waypoint following can produce infinite acceleration at waypoints, motivating smooth trajectories for physical systems. The paper automates cubic-spline construction by assembling interpolation and continuity constraints into a linear system, and illustrates it with free-form curves and signature digitization.
Problem
Piecewise-linear waypoint following changes velocity instantaneously at interior waypoints, implying infinite acceleration that physical actuators cannot produce.
Source: §1 · p. 1Method
The program assembles waypoint interpolation and continuity constraints into a linear system and solves directly for cubic-spline coefficients.
Source: §1 · p. 2, §3.1 · p. 4Results
The program produced smooth free-form curves through hand-placed waypoints and continuous digital signature traces reconstructed from sampled points.
Source: §5.1 · p. 5, §5.2 · p. 5Takeaways & Limitations
The demonstrated drawing and signature tasks use the same spline interpolation machinery described as reusable for feasible motion commands for physical autonomous systems.
Source: §1 · p. 2, §5 · p. 4, §5.1 · p. 5The spline enforces only C2 continuity and does not bound velocity, acceleration, or jerk, so its continuity guarantee alone does not ensure actuator-feasible commands.
Source: §6 · p. 5
Abstract
from arXiv · showhide
We presented a program that automates the construction and solution of the cubic spline interpolation problem for a sequence of position-time waypoints, producing trajectories that are continuous in position, velocity, and acceleration and therefore avoid the infinite-acceleration problem inherent to piecewise-linear waypoint following. By assembling the interpolation and continuity conditions into a single linear system and solving directly for the spline coefficients, the program removes the need to manually derive per-segment equations for each new waypoint set. We illustrated the approach on free-form curve drawing and signature digitization, two tasks that use the same underlying machinery required for smooth robot and drone trajectory generation. Future work includes extending the implementation to multi-axis trajectories with explicit velocity and acceleration bounds, adopting the more computationally efficient tridiagonal solution method for larger waypoint sets, and evaluating the approach quantitatively against alternative trajectory generation methods.
1 Introduction
Piecewise-linear waypoint following creates instantaneous velocity changes and therefore infinite acceleration, while C2 cubic splines preserve position, velocity, and acceleration continuity. The paper automates spline construction and illustrates it with free-form drawing and signature digitization.
1 Introduction
- Piecewise-linear waypoint following can cause instantaneous velocity changes and infinite acceleration, which physical actuators cannot produce.C2 continuity keeps position, velocity, and acceleration continuous; cubic splines are the lowest-degree polynomials that can provide this while interpolating arbitrary waypoints.Source: §1 · pp. 1–2
- Cubic splines guarantee C2 continuity, whereas some aerial-robotics applications use continuity through the fourth derivative, or snap.The higher continuity is used to smooth commanded thrust and body rates; cubic splines are cheaper and sufficient where jerk and higher derivatives are not explicitly constrained.Source: §1 · p. 2
- The program assembles interpolation and continuity constraints into a linear system and solves directly for spline coefficients from position-time waypoints.It is demonstrated on free-form curves and signature traces, using interpolation machinery also relevant to physical autonomous-system trajectories.Source: §1 · p. 2
2 Related Work
The paper situates its direct polynomial-coefficient construction alongside classical spline methods and piecewise-polynomial trajectory generation. It also connects signature reconstruction to prior uses of splines for smoothing and vectorizing signature data.
2 Related Work
- The paper’s formulation follows the direct polynomial-coefficient construction described by de Boor, rather than the B-spline basis; the cited text also covers existence, uniqueness, and optimality of interpolating splines.Source: §2 · p. 2
- Minimum-snap quadrotor trajectories use higher-degree piecewise polynomials to minimize integrated squared snap under waypoint and boundary constraints.The paper shares the pattern of combining interpolation and derivative-continuity constraints in a linear system, but cubic splines require continuity only through the second derivative and do not directly control jerk or snap.Source: §2 · p. 2
- Prior signature-processing work used cubic splines to smooth sampled pen positions [3], while piecewise-cubic Bézier variants were used to vectorize and reconstruct signature contours.These uses are analogous to reconstructing a continuous signature curve from sampled points.Source: §2 · p. 2
3 Problem Formulation
The formulation fits one cubic polynomial to each interval between ordered waypoints, requiring exact waypoint interpolation and C2 continuity. Boundary conditions provide the two additional equations needed to determine all coefficients in a single linear system.
3 Problem Formulation
- The formulation takes n + 1 waypoints with strictly increasing parameters and seeks a spline passing through each waypoint while remaining twice continuously differentiable.The parameter is typically time, but may also be another monotonically increasing quantity such as arc length.Source: §3 · p. 3
- The spline uses n piecewise cubic polynomials, one for each interval between adjacent waypoints.Each segment is expressed using four coefficients in powers of the offset from its left endpoint.Source: §3 · p. 3
- Each segment matches the waypoint values at both endpoints, giving 2n interpolation equations.These conditions enforce S_i(t_i)=x_i and S_i(t_{i+1})=x_{i+1}.Source: §3 · p. 3
- At every interior waypoint, adjoining segments must match in both first and second derivatives, preserving velocity and acceleration continuity.Together these continuity constraints contribute 2(n − 1) equations.Source: §3 · p. 3
- The interpolation and continuity conditions provide 4n − 2 equations for 4n coefficients, so two boundary equations are required.Natural boundaries set endpoint second derivatives to zero; clamped boundaries prescribe endpoint first derivatives, such as a known starting or ending velocity.Source: §3 · p. 3
3.1 Matrix Formulation
- All constraints, including the two boundary conditions, are linear in the unknown coefficients and can be assembled into one matrix equation.The coefficient vector stacks the four coefficients per segment; the sparse matrix is built from waypoint times, and the right-hand side contains waypoint values and boundary-condition constants.Source: §3.1 · pp. 3–4
- The program constructs the matrix and right-hand side from the input waypoint pairs, then solves for all spline coefficients in one step.This avoids deriving the matrix entries by hand for each new waypoint set.Source: §3.1 · p. 4
4 Implementation
The implementation builds a linear system from waypoint and continuity conditions, solves for segment coefficients, and evaluates the resulting spline at query times. Direct inversion is adequate for the waypoint counts used, but the paper identifies a more efficient alternative for larger problems.
4 Implementation
- For n + 1 waypoints, the program constructs a 4n × 4n system, solves for the coefficient vector, and stores the per-segment cubic coefficients.The algorithm fills rows for interpolation, first- and second-derivative continuity, and the selected boundary condition.Source: §4 · p. 4
- Stored coefficients are evaluated at a query time by locating its containing segment and evaluating that segment's cubic polynomial.The supported query interval is from the first to the last waypoint time.Source: §4 · p. 4
- Direct inversion costs O(n^3), while the classical tridiagonal formulation solves for waypoint second derivatives in O(n) time.Direct inversion was not a practical bottleneck for the waypoint counts used; the tridiagonal method is preferable for large waypoint sets or real-time replanning.Source: §4 · p. 4
5 Applications
The program converts sparse samples into smooth continuous curves in two demonstrations: hand-placed free-form paths and reconstructed digital signatures.
5 Applications
- Both applications convert sparse sample points into smooth, continuous curves.The demonstrations use the same spline-fitting program.Source: §5 · p. 4
5.1 Free-form path synthesis
- Hand-placed (x, t) waypoints produce a smooth curve that passes through every waypoint.Source: §5.1 · p. 5
5.2 Signature digitization
- For signature digitization, the program fits a spline to sampled pen positions to create a continuous digital trace from fewer points.The trace is suitable for digital storage or display; prior work used splines to smooth noisy pen-position sequences before further analysis [3].Source: §5.2 · p. 5
6 Discussion and Limitations
The formulation fits each spatial coordinate independently using shared waypoint times, producing a vector trajectory that is C2 in each component. Its continuity guarantees do not bound derivative magnitudes or determine waypoint timing.
6 Discussion and Limitations
- Independent splines for each coordinate, sharing waypoint times, produce a vector trajectory with continuous velocity and acceleration.The formulation fits x(t), y(t), and, for aerial vehicles, z(t); the resulting trajectory is C2 in each component.Source: §6 · p. 5
- C2 continuity does not bound velocity or acceleration magnitudes or constrain jerk, so actuator feasibility is not guaranteed.The passage contrasts direct linear solving with minimum-snap optimization, which incorporates actuator limits at greater computational cost.Source: §6 · p. 5
- Waypoint times are given rather than selected, although time allocation affects whether closely spaced waypoints must be traversed unrealistically quickly.Source: §6 · p. 5
7 Conclusion
The program automates cubic-spline construction and coefficient solving for position-time waypoints, avoiding manual derivation for each waypoint set. It demonstrates the method on curve drawing and signature digitization, with multi-axis bounds, faster solving for larger sets, and quantitative comparisons left for future work.
7 Conclusion
- The program assembles interpolation and continuity conditions into one linear system and directly solves for spline coefficients.This removes the need to derive per-segment equations manually for each waypoint set.Source: §7 · p. 5
- The approach is illustrated through free-form curve drawing and signature digitization, which use the same interpolation machinery as robot and drone trajectory generation.Source: §7 · p. 5
- Future work includes multi-axis trajectories with explicit velocity and acceleration bounds and a tridiagonal solver for larger waypoint sets.Source: §7 · p. 5
- The approach remains to be evaluated quantitatively against alternative trajectory-generation methods.Source: §7 · p. 6