Question 1

What is the physical meaning of each term in M(θ)θ̈+C(θ,θ̇)θ̇+G(θ)=τ?

Accepted Answer

M(θ) is the joint-angle-dependent inertia matrix; multiplied by θ̈ (joint angular acceleration) it gives the torque required to overcome inertia. C(θ,θ̇) is the Coriolis and centrifugal matrix capturing cross-coupling between joint velocities. G(θ) is the gravity vector — the static torque each joint must exert to maintain posture against gravity. The sum of these three terms equals the required joint torque τ, which sets the motor design target.

Question 2

What is a cubic-spline trajectory?

Accepted Answer

A cubic-spline trajectory is a third-order polynomial θ(t)=a₀+a₁t+a₂t²+a₃t³ that satisfies position and zero-velocity boundary conditions at start and end (rest-to-rest motion). The velocity θ̇ and acceleration θ̈ remain smooth and continuous, reducing mechanical shock and motor overload. Widely used in point-to-point (PTP) industrial robot control.

Question 3

How do you select a motor for a robot arm?

Accepted Answer

Basic procedure: ① compute peak joint torque τ_peak from trajectory analysis; ② calculate continuous RMS torque τ_rms for sustained operation; ③ choose gear ratio N and motor rated torque T_motor such that N×T_motor ≥ τ_peak; ④ verify that the inertia ratio (reflected load inertia / motor rotor inertia) is roughly 1:1 to 1:5; ⑤ confirm maximum speed and acceleration limits are satisfied. Use the peak torque values from this tool to compare against motor datasheets.

Question 4

What is the difference between Newton-Euler and Lagrangian methods?

Accepted Answer

Newton-Euler computes forces and moments on each link in two recursive passes (base-to-tip forward pass, tip-to-base backward pass). It is computationally efficient — O(n) per time step — and well suited for high-DOF robots. The Lagrangian method derives equations of motion from kinetic and potential energy and yields symbolic, closed-form equations convenient for theoretical analysis. CAE multibody tools such as Adams typically use a Lagrangian (generalized-coordinate) formulation.

Robot Joint Torque & Newton-Euler Dynamics Calculator

Equations of Motion (2-DOF)