Cubic Trajectory Polynomial Calculator

Endpoint position/velocity cubic polynomial and bounded sampled motion, coefficient order ascending; no feasible acceleration/actuator limits implied. Caller independently establishes units, regime, calibration and applicability; this does not establish design-code compliance, authenticity, operational safety, treatment efficacy or a validated forecast.

Description

A cubic trajectory can meet specified positions and velocities at two endpoint times.

A cubic trajectory can meet specified positions and velocities at two endpoint times.^[1]

Meeting those constraints does not independently limit acceleration, jerk or actuator demand. Use this tool to calculate cubic Trajectory Polynomial from the supplied measurements and assumptions.

When to use Cubic Trajectory Polynomial Calculator

Use this tool to calculate cubic Trajectory Polynomial from the supplied measurements and assumptions. Establish the SI units or caller unit basis, load signs, measurement interval, model conditions and independently chosen coefficients or references.

How to use Cubic Trajectory Polynomial Calculator and understand its results

Establish the SI units or caller unit basis, load signs, measurement interval, model conditions and independently chosen coefficients or references.
Enter initialPosition, finalPosition, initialVelocityPerSecond, finalVelocityPerSecond, durationSeconds, queryElapsedSeconds, using the field meanings and rules below, then run the tool.
Read cubicCoefficients, samplePositions, sampleVelocitiesPerSecond. Interpret the fields using their units, signs and evidence conventions below.
If input validation fails, correct the reported field and run again. Before applying the result, check reference scope, model assumptions and any missing evidence or placeholder values.

Input field	Meaning	Value type	Requirement
initialPosition	Initial position	number	Required
finalPosition	Final position	number	Required
initialVelocityPerSecond	Initial velocity per second	number	Required
finalVelocityPerSecond	Final velocity per second	number	Required
durationSeconds	Duration seconds	number	Required
queryElapsedSeconds	Query elapsed seconds	list of number	Required

Enter finite endpoint positions and velocities, positive total time and query times within inclusive zero..totalTime. Solve the cubic coefficients from endpoint conditions and evaluate position and velocity at each query. Preserve query order; no velocity or acceleration limit is selected.

x(t)=x0+v0t+a2t²+a3t³; a2=3(x1−x0)/T²−(2v0+v1)/T; a3=−2(x1−x0)/T³+(v0+v1)/T²

Use the exact units, boundary rules, ordering and missing-evidence conventions above.

Output field	Meaning	Value type
cubicCoefficients	Cubic Coefficients; follow the units, sign, ordering and evidence convention stated above	list of number
samplePositions	Sample Positions; follow the units, sign, ordering and evidence convention stated above	list of number
sampleVelocitiesPerSecond	Sample Velocities Per Second; follow the units, sign, ordering and evidence convention stated above	list of number

json

{
  "input": {
    "initialPosition": 0,
    "finalPosition": 1,
    "initialVelocityPerSecond": 0,
    "finalVelocityPerSecond": 0,
    "durationSeconds": 2,
    "queryElapsedSeconds": [
      0,
      1,
      2
    ]
  },
  "output": {
    "cubicCoefficients": [
      0,
      0,
      0.75,
      -0.25
    ],
    "samplePositions": [
      0,
      0.5,
      1
    ],
    "sampleVelocitiesPerSecond": [
      0,
      0.75,
      0
    ]
  }
}

Limitations and assumptions

Meeting those constraints does not independently limit acceleration, jerk or actuator demand. Interpretation depends on the units, sampling scope and exact model described above.

All listed fields require explicit values. Unexpected fields, incorrect types, nonfinite numbers and sparse evidence arrays fail validation; the specific bounds and alignment requirements are stated above. Extremely large inputs or outputs can also be rejected. An input error is distinct from a successful result reporting false, no matches or missing evidence.

Alternative or Complementary methods

Use 2R Robot Forward Kinematics Calculator for this related task: Planar two-link forward kinematics maps joint angles and link lengths to a position.

Use 2R Robot Inverse Kinematics Calculator when its separate inputs and assumptions fit the question: A planar two-link arm can reach a point through two angular branches when geometry permits.

References

Cubic Trajectory Polynomial Calculator — tool.io

Similar or alternative tools

2R Robot Forward Kinematics Calculator
Planar two-link forward kinematics maps joint angles and link lengths to a position.
2R Robot Inverse Kinematics Calculator
A planar two-link arm can reach a point through two angular branches when geometry permits.