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Mathematical Notation

This page defines the mathematical notation, symbols, and conventions used consistently throughout the textbook.

Constitution Requirement

This notation guide ensures consistency across all chapters, as mandated by Constitution Principle III (Consistency & Standards).

General Conventions​

Scalars: Lowercase italic (x, t, θ) Vectors: Lowercase bold (v, p, q) Matrices: Uppercase bold (R, T, J) Sets: Uppercase calligraphic

Robotics Notation​

Kinematics​

  • q: Joint positions vector (rad or m)
  • q̇: Joint velocities vector (rad/s or m/s)
  • q̈: Joint accelerations vector (rad/s² or m/s²)
  • p: End-effector position (m)
  • R: Rotation matrix 3×3
  • T: Homogeneous transformation 4×4
  • J: Jacobian matrix

Dynamics​

  • τ: Joint torque (N⋅m)
  • F: Force (N)
  • m: Mass (kg)
  • I: Moment of inertia (kg⋅m²)
  • g: Gravitational acceleration (m/s², typically 9.81)

Sensors​

  • ω: Angular velocity (rad/s)
  • α: Angular acceleration (rad/s²)
  • a: Linear acceleration (m/s²)
  • d: Distance measurement (m)

Angle Representations​

Euler Angles (ZYX Convention)​

  • φ (phi): Roll, rotation about X-axis, range [-π, π] rad
  • θ (theta): Pitch, rotation about Y-axis, range [-π/2, π/2] rad
  • ψ (psi): Yaw, rotation about Z-axis, range [-π, π] rad

Quaternions​

Representation: q = [qw, qx, qy, qz]

  • qw: scalar part
  • qx, qy, qz: vector part
  • Constraint: qw² + qx² + qy² + qz² = 1

Units (SI Preferred)​

  • Length: meter (m)
  • Mass: kilogram (kg)
  • Time: second (s)
  • Force: Newton (N)
  • Torque: Newton-meter (N⋅m)
  • Angle: radian (rad)
  • Angular velocity: rad/s
  • Frequency: Hertz (Hz)

Domain-Specific Exceptions​

  • Joint angles may be specified in degrees (noted explicitly, e.g., "45°")
  • Motor speeds may use RPM (standard motor specification unit)

Control Theory​

  • e(t): Error signal
  • Kp: Proportional gain
  • Ki: Integral gain
  • Kd: Derivative gain
  • G(s): Transfer function

Example Usage​

Forward Kinematics:

T(base→ee) = T₁(q₁) · T₂(q₂) · ... · Tₙ(qₙ)

PID Control:

τ(t) = Kp·e(t) + Ki·∫e(τ)dτ + Kd·de(t)/dt

Jacobian Relationship:

v = J(q) · q̇
Consistency

When you see these symbols in equations throughout the textbook, refer back to this page for their precise definitions and units.