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Qwestrum Engineering360 · Mechanical Engineering · Robotics

Trajectory Planning

Trajectory planning generates smooth joint (or Cartesian) paths versus time; a cubic polynomial q(t) = a₀ + a₁t + a₂t² + a₃t³ meets position and velocity boundary conditions, per robotics texts.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Joint space vs Cartesian space planning
  • Trapezoidal velocity profile: accel, cruise, decel
  • Jerk limitation reduces vibration and wear

Topic details

Introduction

Trajectory planning produces the time history of motion between waypoints so the robot moves smoothly without exceeding velocity, acceleration, or jerk limits. Indian robotics courses cover polynomial and trapezoidal-velocity trajectories.

Scope in B.Tech and GATE syllabus

A cubic polynomial satisfies four boundary conditions (start/end position and velocity), giving smooth position and velocity but a discontinuous acceleration at the ends. Quintic polynomials add acceleration continuity; trapezoidal-velocity profiles limit acceleration explicitly.

Why this topic matters in practice

Planning can be in joint space (simple, avoids singularities) or Cartesian space (straight-line tool paths). Deriving polynomial coefficients from boundary conditions is the standard exam task.

Key relations & formulas

q(t)=a0+a1t+a2t2+a3t3q(t) = a_{0} + a_{1}t + a_{2}t^{2} + a_{3}t^{3}
(cubic polynomial joint trajectory)

Formulas (Indian textbook notation)

  • q˙max,q¨maxconstraintslimitmotiontimeq̇_max, q̈_max constraints limit motion time

Formulas (Indian textbook notation)

  • LinearCartesianpath:interpolateposition,solveIKateachpointLinear Cartesian path: interpolate position, solve IK at each point

Formulas (Indian textbook notation)

  • BlendradiusratviapointsforsmoothmotionBlend radius r at via points for smooth motion

Notation and sign conventions

Relation 1 —
qq
q(t)=a0+a1t+a2t2+a3t3q(t) = a_{0} + a_{1}t + a_{2}t^{2} + a_{3}t^{3}
(cubic polynomial joint trajectory)
Write this relation with symbols exactly as in Robotics & Control — Nagrath & Ghosh before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
q˙max,q¨maxconstraintslimitmotiontimeq̇_max, q̈_max constraints limit motion time

Formulas (Indian textbook notation)

  • q˙max,q¨maxconstraintslimitmotiontimeq̇_max, q̈_max constraints limit motion time
Write this relation with symbols exactly as in Robotics & Control — Nagrath & Ghosh before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
LinearCartesianpath:interpolateposition,solveIKateachpointLinear Cartesian path: interpolate position, solve IK at each point

Formulas (Indian textbook notation)

  • LinearCartesianpath:interpolateposition,solveIKateachpointLinear Cartesian path: interpolate position, solve IK at each point
Write this relation with symbols exactly as in Robotics & Control — Nagrath & Ghosh before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
BlendradiusratviapointsforsmoothmotionBlend radius r at via points for smooth motion

Formulas (Indian textbook notation)

  • BlendradiusratviapointsforsmoothmotionBlend radius r at via points for smooth motion
Write this relation with symbols exactly as in Robotics & Control — Nagrath & Ghosh before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

The simplest smooth point-to-point motion uses a cubic q(t) = a₀ + a₁t + a₂t² + a₃t³. The four coefficients are found from the boundary conditions on position and velocity at the start and end, giving a smooth position and a parabolic velocity.

Governing relations in practice

A cubic has a step in acceleration at the endpoints (finite jerk spike); a quintic (fifth-order) polynomial adds acceleration boundary conditions for a fully smooth motion, used where jerk must be limited.

Design and analysis considerations

Trapezoidal velocity profiles (linear segments with parabolic blends) accelerate at a constant limit, cruise at maximum velocity, then decelerate — time-optimal within acceleration limits and common in industrial controllers.

Advanced theory and extensions

Joint-space planning interpolates each joint independently (simple, guaranteed within limits) but the tool path is not a straight line; Cartesian-space planning gives predictable tool paths but must check for singularities and reachability. Choosing the profile and deriving its coefficients is the applied skill.

Assumptions and validity limits

State assumptions explicitly before using any relation for trajectory planning — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Robotics viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Robotics papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to trajectory planning.
4. Use equation 1:
qq
.
5. Use equation 2:
q˙max,q¨maxconstraintslimitmotiontimeq̇_max, q̈_max constraints limit motion time
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Trajectory Planning appears in industrial automation and research labs. In Indian mechanical curricula this topic is tested because it connects theory to robot kinematics, sensing, and control.
GATE and semester exams often combine trajectory planning with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use trajectory planning?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using a cubic where acceleration continuity (quintic) is required
• Forgetting to apply all boundary conditions when solving for coefficients
• Confusing joint-space (simple) with Cartesian-space (straight-line) planning
• Ignoring velocity/acceleration limits so the trajectory is infeasible

Quick revision checklist

Before attempting trajectory planning problems, confirm you can:
1. Joint space vs Cartesian space planning
2. Trapezoidal velocity profile: accel, cruise, decel
3. Jerk limitation reduces vibration and wear
Revise the solved examples in Robotics & Control — Nagrath & Ghosh and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Cubic trajectory coefficients

Problem

A joint moves from 0° to 90° in 2 s, starting and ending at rest, via a cubic. Find the coefficients a₀ and a₂.

Solution

With q(0)=0, q̇(0)=0: a₀ = 0, a₁ = 0. Then a₂ = 3(q_f)/t² = 3×90/2² = 67.5 (and a₃ = −2×90/2³ = −22.5).

Conceptual check — Trajectory Planning

Problem

In a Robotics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of trajectory planning." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Trajectory Planning, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Trajectory planning generates smooth joint (or Cartesian) paths versus time; a cubic polynomial q(t) = a₀ + a₁t + a₂t² + a₃t³ meets position and velocity boundary conditions, per robotics texts.
  2. 2
    State the relation q and name each symbol.

    Model answer

    The governing relation is qq. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation q̇_max, q̈_max constraints limit motion time and name each symbol.

    Model answer

    The governing relation is q˙max,q¨maxconstraintslimitmotiontimeq̇_max, q̈_max constraints limit motion time. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation Linear Cartesian path: interpolate position, solve IK at each point and name each symbol.

    Model answer

    The governing relation is LinearCartesianpath:interpolateposition,solveIKateachpointLinear Cartesian path: interpolate position, solve IK at each point. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation Blend radius r at via points for smooth motion and name each symbol.

    Model answer

    The governing relation is BlendradiusratviapointsforsmoothmotionBlend radius r at via points for smooth motion. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Joint space vs Cartesian space planning

    Model answer

    Joint space vs Cartesian space planning — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Trapezoidal velocity profile: accel, cruise, decel

    Model answer

    Trapezoidal velocity profile: accel, cruise, decel — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Jerk limitation reduces vibration and wear

    Model answer

    Jerk limitation reduces vibration and wear — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Using a cubic where acceleration continuity (quintic) is required?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Forgetting to apply all boundary conditions when solving for coefficients?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Confusing joint-space (simple) with Cartesian-space (straight-line) planning?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Ignoring velocity/acceleration limits so the trajectory is infeasible?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Nagrath & Ghosh Ch. 5 — boundary conditions fix cubic coefficients.
  • 2
    Avoid: Using a cubic where acceleration continuity (quintic) is required
  • 3
    Avoid: Forgetting to apply all boundary conditions when solving for coefficients
  • 4
    Avoid: Confusing joint-space (simple) with Cartesian-space (straight-line) planning

📖 Standard books (India)

  • Robotics & Control — Nagrath & Ghosh

    Read: Syllabus unit

    Kinematics, sensors, and industrial robots

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