Questions: The Robot Jacobian and Velocity Control

Question 1

For a 2-link planar robot arm with forward kinematics x = L₁·cos(θ₁) + L₂·cos(θ₁ + θ₂), y = L₁·sin(θ₁) + L₂·sin(θ₁ + θ₂), the Jacobian J relates joint velocities [θ̇₁, θ̇₂]ᵀ to end-effector velocity [ẋ, ẏ]ᵀ. Which is the correct expression for J?

• A. J = [[-L₁·sin(θ₁) - L₂·sin(θ₁ + θ₂), -L₂·sin(θ₁ + θ₂)], [L₁·cos(θ₁) + L₂·cos(θ₁ + θ₂), L₂·cos(θ₁ + θ₂)]]
• B. J = [[L₁·cos(θ₁), L₂·cos(θ₁ + θ₂)], [L₁·sin(θ₁), L₂·sin(θ₁ + θ₂)]]
• C. J = [∂x/∂θ₁, ∂x/∂θ₂; ∂y/∂θ₁, ∂y/∂θ₂] = [[-L₁·sin(θ₁) - L₂·sin(θ₁ + θ₂), -L₂·sin(θ₁ + θ₂)], [L₁·cos(θ₁) + L₂·cos(θ₁ + θ₂), L₂·cos(θ₁ + θ₂)]]
• D. Options (a) and (c) are equivalent

Answer: (D)

Near-singular configurations are almost as problematic as true singularities from a practical control perspective. This is why motion planners include singularity-avoidance heuristics: they maintain the Jacobian well-conditioned by modifying reference trajectories to avoid configurations with high condition numbers.

Question 2

A 6-DOF robot arm is commanded with a desired end-effector velocity [v_x, v_y, v_z, ω_x, ω_y, ω_z]ᵀ (3 linear + 3 angular velocity components). The Jacobian is 6×6. If this Jacobian is singular, which of the following must be true?

• T. True
• F. False

Answer: True

At least one of the six velocity components cannot be achieved, or at least one joint would require infinite velocity. The rank of J must be less than 6, meaning the null space is non-empty and there exist desired velocities that lie outside the range of J.