Velocity kinematics and the Jacobian

Why it matters

Welding, gluing and painting need the tool to move along a path at a controlled speed, and the controller can only command joint speeds. The Jacobian is the matrix that converts between the two. It also reveals singularities, maps tool forces to joint torques and drives numerical inverse kinematics, so it is the single most useful object in manipulator analysis after the transform itself.

Key ideas

Definition. Differentiate forward kinematics x = f(q) with respect to time:

ẋ = J(q)·q̇, where J = ∂f/∂q.

• q̇ — joint velocities (rad/s for revolute joints, m/s for prismatic).
• ẋ — tool velocity in task space: linear velocity v (m/s) and, in the full form, angular velocity ω (rad/s).
• J is configuration dependent: it changes as the arm moves.

Size. For n joints and an m-dimensional task, J is m × n. The full geometric Jacobian of a spatial arm is 6 × n (3 rows for v, 3 for ω). A planar arm's position-only Jacobian is 2 × n; adding the planar rotation makes it 3 × n.

Analytical vs geometric Jacobian. The analytical Jacobian differentiates a chosen orientation representation (e.g. Euler angles), so its lower rows give Euler-angle rates; the geometric Jacobian gives true angular velocity ω. They are related by a matrix that depends on the angles.

Building the geometric Jacobian column by column (all vectors in the base frame, z_{i−1} is joint i's axis, o_{i−1} its origin, oₙ the tool point):

• revolute joint i: Jᵥᵢ = z_{i−1} × (oₙ − o_{i−1}), J_ωᵢ = z_{i−1};
• prismatic joint i: Jᵥᵢ = z_{i−1}, J_ωᵢ = 0. Physically, column i is the tool velocity produced by joint i moving at unit speed with all others locked.

Inverse velocity problem. For a square, non-singular J: q̇ = J⁻¹·ẋ. For a redundant arm (n > m) use the pseudo-inverse q̇ = J⁺·ẋ = Jᵀ(JJᵀ)⁻¹·ẋ, which gives the minimum-norm joint velocity. Near singularities use damped least squares.

Singularity. When J loses rank (det J = 0 for square J) the tool cannot move in some direction, and finite tool speeds in nearby directions demand very large joint speeds. Detailed in the next topic.

Statics link. By virtual work, the same matrix maps tool forces to joint torques: τ = Jᵀ·F. This duality is why the Jacobian appears in force control.

Acceleration. Differentiating again: ẍ = J·q̈ + J̇·q̇ — used in Cartesian trajectory tracking.

Formulas

ẋ = J(q) · q̇

q̇ = J⁻¹ · ẋ (square, non-singular J); q̇ = J⁺ · ẋ, J⁺ = Jᵀ(JJᵀ)⁻¹ (redundant arm)

Planar 2R (position rows):

J = [[−L₁·sin θ₁ − L₂·sin(θ₁ + θ₂), −L₂·sin(θ₁ + θ₂)], [L₁·cos θ₁ + L₂·cos(θ₁ + θ₂), L₂·cos(θ₁ + θ₂)]]

• L₁, L₂ in m; entries in m/rad, so J·q̇ is in m/s.

det J = L₁ · L₂ · sin θ₂ (m²) — zero when θ₂ = 0° or 180°.

ω_tool = θ̇₁ + θ̇₂ (planar tool angular velocity, rad/s)

Revolute column: Jᵥᵢ = z_{i−1} × (oₙ − o_{i−1}), J_ωᵢ = z_{i−1}; prismatic: Jᵥᵢ = z_{i−1}, J_ωᵢ = 0

τ = Jᵀ · F

Worked examples

Example 1 (standard). Planar 2R, L₁ = L₂ = 1 m, θ₁ = 30°, θ₂ = 45°, θ̇₁ = 1 rad/s, θ̇₂ = 0.5 rad/s. Find the tool velocity.

1. sin 30° = 0.5, cos 30° = 0.8660; θ₁ + θ₂ = 75°: sin 75° = 0.9659, cos 75° = 0.2588.
2. J = [[−0.5 − 0.9659, −0.9659], [0.8660 + 0.2588, 0.2588]] = [[−1.4659, −0.9659], [1.1248, 0.2588]] m/rad.
3. ẋ = −1.4659·1 + (−0.9659)·0.5 = −1.4659 − 0.4830 = −1.9489 m/s.
4. ẏ = 1.1248·1 + 0.2588·0.5 = 1.1248 + 0.1294 = 1.2542 m/s.
5. Speed = √(1.9489² + 1.2542²) = 2.318 m/s; tool angular velocity = 1.5 rad/s.

Answer: v = (−1.949, 1.254) m/s, |v| = 2.32 m/s.

Example 2 (GATE level). Planar 2R, L₁ = 0.5 m, L₂ = 0.3 m, at θ₁ = 0°, θ₂ = 90°. The tool must move at ẋ = 0.1 m/s, ẏ = 0.2 m/s. Find the joint rates.

1. sin θ₁ = 0, cos θ₁ = 1, sin(θ₁ + θ₂) = 1, cos(θ₁ + θ₂) = 0.
2. J = [[−0 − 0.3, −0.3], [0.5 + 0, 0]] = [[−0.3, −0.3], [0.5, 0]].
3. det J = (−0.3)(0) − (−0.3)(0.5) = 0.15 m²; check L₁L₂ sin 90° = 0.15 ✓ (non-singular).
4. Row 2: 0.5·θ̇₁ = 0.2 → θ̇₁ = 0.4 rad/s.
5. Row 1: −0.3·0.4 − 0.3·θ̇₂ = 0.1 → θ̇₂ = (0.1 + 0.12)/(−0.3) = −0.733 rad/s.

Answer: θ̇₁ = 0.400 rad/s, θ̇₂ = −0.733 rad/s.

Common mistakes

• Treating the Jacobian as constant — it must be evaluated at the current q.
• Using θ₂ instead of θ₁ + θ₂ in the second column.
• Giving J for a spatial arm as n × n — the geometric Jacobian is 6 × n.
• Inverting a singular or non-square J; use the pseudo-inverse or damped least squares.
• Mixing degrees and rad/s: joint rates must be in rad/s for v in m/s.
• Confusing analytical-Jacobian Euler-angle rates with true angular velocity.

For GATE ME

Expect a planar 2R or 3R arm with given angles and joint rates: compute the tool velocity, the joint rates for a required tool velocity, or det J and the singular angles. Conceptual questions test the size of J, its rank at singularities and the transpose relation τ = Jᵀ·F. Practise writing the 2R Jacobian from memory and solving 2 × 2 systems quickly.

Quick check

1. What is the size of the geometric Jacobian of a 7-joint spatial arm?
2. Planar 2R with L₁ = 0.4 m, L₂ = 0.5 m, θ₂ = 30°: det J?
3. What does column i of J represent?
4. For which θ₂ is a planar 2R arm singular?
5. Write the relation between tool force and joint torque.

Answers: 1. 6 × 7; 2. 0.1 m²; 3. Tool velocity produced by unit speed of joint i alone; 4. θ₂ = 0° or 180°; 5. τ = Jᵀ·F.