Moving a robot smoothly: why average speed is not enough

The problem: arriving on time does not describe the journey

A robotic axis carries a tray between positions 20 centimetres apart in one second. Its average speed is 0.2 metres per second. That seems enough to compare two programs, but hides how the motor accelerates and brakes. Commands with equal distance and duration can demand very different efforts. Which quantities must we check so an apparently slow movement does not require excessively abrupt changes?

Abstract. We construct three position laws, derive velocity, acceleration and jerk, then calculate how duration must increase to satisfy specified limits. A smoother start can increase some middle-of-motion peaks. These are kinematic calculations with educational inputs: we have not measured robot vibration, force, consumption or safety. We will distinguish a mathematically admissible profile from motion validated on a machine.

Before equations: three kinds of change

Position says where the tray is. Velocity measures position change per second; acceleration measures velocity change per second. Jerk is the rate of change of acceleration. Units are metres, m/s, m/s² and m/s³. In a car we may feel a sustained push while accelerating and a jolt when that push changes abruptly. This is useful intuition, not a model of the robot structure.

Assume one linear axis, a straight path, rest at both ends, and constant position before and after motion. We do not model elasticity, friction, payload, control or contact. Separate geometry—travelling D=0.20 m—from timing. Let u=t/T be dimensionless normalized time from zero to one. The function p(u) gives the completed path fraction. Our model is x(t)=D p(t/T), with the origin at the initial position.

To understand duration, differentiate with respect to time. Each derivative introduces 1/T. A prime denotes differentiation with respect to u, not a physical velocity.

For equal shape and distance, doubling T halves velocities, quarters accelerations and divides jerk by eight. It does not halve every demand. This is a planning lever, but does not establish structural vibration, which requires dynamics and measurements. We can now compare shapes at fixed distance and units.

The cubic: the problem hides at the boundaries

Require p(0)=0, p(1)=1 and zero endpoint velocities. A cubic has four coefficients for four conditions. Write p=b₀+b₁u+b₂u²+b₃u³. Initial conditions give b₀=b₁=0; final conditions give b₂+b₃=1 and 2b₂+3b₃=0. Hence b₂=3 and b₃=−2. The curve comes from motion requirements, not appearance.

For D=0.20 m and T=1 s, peak velocity is 0.300 m/s, peak absolute acceleration 1.200 m/s² and interior jerk −2.400 m/s³. This does not mean the entire motion satisfies 5 m/s³: acceleration jumps from zero before starting to 1.200. That jump has no finite ordinary derivative; another occurs at the end. Increasing duration reduces its magnitude but does not make it continuous.

The quintic: removing a jump has a price

Add zero acceleration at both ends: six conditions suggest a fifth-degree polynomial. The first three coefficients vanish; the others solve b₃+b₄+b₅=1, 3b₃+4b₄+5b₅=0 and 6b₃+12b₄+20b₅=0. The solution is 10, −15, 6. We added acceleration continuity with rest, not a vague promise of perfect motion.

The peak of p₅′ is 1.875 at mid-time. Extrema of p₅″ satisfy p₅‴=0: u=(3±√3)/6, giving maximum magnitude 10/√3. Peak absolute p₅‴ is 60 at the boundaries. Physical scaling gives 0.375 m/s, 1.155 m/s² and 12.000 m/s³. Initial acceleration is zero, but jerk jumps from zero to 12. This finite jerk jump differs from an acceleration jump. Jerk is bounded almost everywhere, although there is no unique classical value at the join.

A third profile to challenge intuition

Also require zero endpoint jerk. Eight conditions yield a seventh-degree polynomial. Substituting u=0 and u=1 into its first three derivatives checks the joins. This is an algebraic comparison, not a new algorithm. The program locates extrema through derivative roots rather than taking the largest of a few samples.

Normalized peaks are 2.1875, about 7.5132 and 52.5 for velocity, acceleration and jerk. Our physical peaks are 0.4375 m/s, 1.5026 m/s² and 10.5 m/s³. Endpoint jerk is zero, but velocity and acceleration peaks exceed the quintic. Gentler boundaries leave less effective time for the same distance; the middle compensates. “Smoother” does not specify which limit improves.

The asterisk matters: the cubic value excludes acceleration jumps at rest joins and is not a finite global jerk maximum. All average speeds remain 0.20 m/s. Inspect boundary acceleration first, then interior peaks. Both distinguish visual appeal from machine feasibility.

How long must the motion take?

Set hypothetical limits V=0.4 m/s, A=1 m/s² and J=5 m/s³, not product specifications. Let M₁, M₂ and M₃ be absolute normalized derivative maxima. Each constraint gives a minimum duration. Choose the largest: satisfying two cannot excuse violating the third. This is for a fixed shape and single axis, not global optimization of all motions.

Quintic bounds are 0.9375 s, 1.0746 s and 1.3389 s: jerk dominates. Degree-seven bounds are 1.0938 s, 1.2258 s and 1.2806 s: jerk dominates again, with slightly lower required duration. This is not a universal ranking. A tighter acceleration limit could disadvantage degree seven. The cubic cannot meet the global jerk constraint at finite T with these joins; using only interior jerk would falsely approve it.

Peak, integral cost and vibration are not synonyms

Another comparison integrates squared jerk over time. This mathematical cost penalizes rapid changes throughout motion, not only the worst instant. Substituting j=(D/T³)p‴ and dt=T du gives I=(D²/T⁵)∫₀¹[p‴(u)]² du. Exact polynomial integrals are 720 for the quintic and 1120 for degree seven, yielding 28.8 and 44.8 m²/s⁵. Continuous endpoint jerk thus has a higher integral cost at equal duration.

These units are not joules: we calculated neither electrical nor mechanical energy. I is not direct vibration measurement either. A flexible payload responds to command frequencies and natural modes; estimating this requires stiffness, damping and control. A useful criterion is not proof of everything. State the optimized property, imposed constraints and excluded phenomena.

Alternatives and the transition to hardware

Trapezoidal velocity profiles are simple alternatives, but instantaneous acceleration changes retain join issues. S-curves organize controlled-jerk, constant-acceleration and constant-velocity phases; they need not equal our polynomials. Multi-joint robots require Cartesian trajectories to be mapped into joint motion and checked for torques, singularities, collisions and per-axis limits. One scalar inequality cannot replace this.

Reproducible code contains coefficients, parameters, versions and results. No seed is needed without randomness. It verifies endpoint positions and velocities, extrema and duration. The snippet below reconstructs the quintic case: maximum selects the dominant constraint. A real test should record commanded and measured position, controller period, saturation, payload, tracking error and settling. This protocol is proposed, not executed.

Appendix: checking degree seven without trusting the plot

To reconstruct degree-seven coefficients, impose endpoint values of p and its first three derivatives. Left conditions leave b₄, b₅, b₆, b₇. Right conditions are b₄+b₅+b₆+b₇=1; 4b₄+5b₅+6b₆+7b₇=0; 12b₄+20b₅+30b₆+42b₇=0; 24b₄+60b₅+120b₆+210b₇=0. Solutions are 35, −84, 70, −20. Each equation means final position, stopping, zero acceleration or zero jerk.

Why roots? An interior extremum of a differentiable function has zero derivative. The program evaluates boundaries and real interior roots of the next derivative, then takes the largest magnitude. Degree-seven velocity peaks at u=1/2; nonzero acceleration extrema occur at u=(5±√5)/10. This is stronger than a coarse grid that could miss peaks. Floating-point arithmetic and explicit code tolerances remain: this is not an automatically formally verified proof.

What we have learned

Moving the tray needs more than distance divided by time. Check how acceleration and jerk join rest, their peaks and duration against all limits. One second is insufficient for our two regular profiles: about 1.339 and 1.281 seconds are required. These answer a kinematic question, not machine certification. Industrial and collaborative robotics is a direction EL-AI intends to explore; this analysis documents no EL-AI robot, installation or company experiment.

Sources and scope

Our reference is chapter 9 of Lynch and Park, Modern Robotics, first edition 2017, updated preprint 30 December 2019, sections 9.2.2.1–9.2.2.3, and the authors’ teaching explanation. The 20 cm move, limits and degree-seven comparison are reproducible educational analysis, not book experiments or peer-reviewed original research.

Lynch & Park — Modern Robotics, chapter 9 (2019 updated preprint).

Lynch & Park — Point-to-Point Trajectories, Part 2.

from math import sqrt
D, V, A, J = 0.2, 0.4, 1.0, 5.0
bounds = (1.875*D/V, sqrt((10/sqrt(3))*D/A), (60*D/J)**(1/3))
print("quintic duration bounds (s):", bounds)
print("minimum within this family (s):", max(bounds))
# Kinematic calculation, not a hardware measurement.

Code, data, and instructions · JSON. Educational calculations executed with Python 3.14.0; figures with Matplotlib 3.11.2. AI-assisted analysis, without claiming peer review or human review. Original illustrative ImageGen cover: it does not document EL-AI people, premises, or installations. Sources accessed 28 September 2026.