Forward kinematics of serial manipulators

Why it matters

Forward kinematics answers the question the controller asks thousands of times a second: given the joint readings, where is the tool and which way is it pointing? It is used to display the tool position on the teach pendant, to check workspace and safety zones, and as the starting point for the Jacobian, inverse kinematics and dynamics.

Key ideas

Definition. For a serial manipulator with joint variables q = (q₁, …, qₙ) — angles θ for revolute joints, displacements d for prismatic joints — forward kinematics (FK) is the map q → ⁰Tₙ(q), the pose (position + orientation) of the tool frame in the base frame. FK always has a unique answer, unlike inverse kinematics.

Procedure.

1. Assign link frames (usually with the DH convention) and write the DH table.
2. Form each link transform Aᵢ(qᵢ).
3. Multiply: ⁰Tₙ = A₁·A₂·…·Aₙ. Add a fixed base transform (robot mounted on a pedestal) and tool transform (gripper length) if needed: ʷT_tool = ʷT₀·⁰Tₙ·ⁿT_tool.
4. Read position from the fourth column and orientation from the 3 × 3 block; convert to roll-pitch-yaw or another set if required.

Planar arms can be solved by geometry directly: each link adds a vector of length Lᵢ at the cumulative angle θ₁ + … + θᵢ. The tool orientation in the plane is φ = θ₁ + θ₂ + … + θₙ.

Articulated (RRR) arm. For waist θ₁, shoulder θ₂ and elbow θ₃ (shoulder height d₁, upper arm a₂, forearm a₃), the arm acts as a planar 2R arm in a vertical plane that is rotated by θ₁ about the vertical axis. This decoupling is the key to both FK and IK of most 6-axis industrial arms; the last three joints form a spherical wrist that only changes orientation.

Joint space vs task space. q lives in joint space (n-dimensional); the tool pose lives in task (Cartesian) space (up to 6-dimensional). FK maps joint space to task space; it is nonlinear because of the sines and cosines.

Error sensitivity. A small joint angle error δθ at a joint produces a tool position error of about r·δθ, where r is the distance from that joint axis to the tool — base joints matter most. This is why encoder resolution and link deflection on the first axes dominate accuracy.

Formulas

⁰Tₙ = A₁(q₁) · A₂(q₂) · … · Aₙ(qₙ)

Planar 2R: x = L₁·cos θ₁ + L₂·cos(θ₁ + θ₂), y = L₁·sin θ₁ + L₂·sin(θ₁ + θ₂), φ = θ₁ + θ₂

Planar 3R: x = L₁·c₁ + L₂·c₁₂ + L₃·c₁₂₃, y = L₁·s₁ + L₂·s₁₂ + L₃·s₁₂₃, φ = θ₁ + θ₂ + θ₃

• cᵢⱼ = cos(θᵢ + θⱼ), sᵢⱼ = sin(θᵢ + θⱼ); Lᵢ in m; angles in degrees or rad.

Articulated RRR: r = a₂·cos θ₂ + a₃·cos(θ₂ + θ₃), x = r·cos θ₁, y = r·sin θ₁, z = d₁ + a₂·sin θ₂ + a₃·sin(θ₂ + θ₃)

• d₁ — shoulder height (m); a₂, a₃ — upper-arm and forearm lengths (m); θ₂ measured from the horizontal.

Position error ≈ r · δθ

• r — distance from the joint axis to the tool (m); δθ — joint error (rad).

Worked examples

Example 1 (standard). A planar 2R arm has L₁ = L₂ = 1 m, θ₁ = 45°, θ₂ = 30°. Find the tool position and orientation.

1. x = L₁·cos θ₁ + L₂·cos(θ₁ + θ₂) = cos 45° + cos 75° = 0.7071 + 0.2588 = 0.9659 m.
2. y = sin 45° + sin 75° = 0.7071 + 0.9659 = 1.6730 m.
3. φ = 45° + 30° = 75°.

Answer: (0.966 m, 1.673 m), φ = 75°.

Example 2 (GATE level). An articulated RRR arm has d₁ = 0.5 m, a₂ = 0.4 m, a₃ = 0.3 m. For θ₁ = 60°, θ₂ = 30°, θ₃ = −60°, find the tool position.

1. Horizontal reach: r = 0.4·cos 30° + 0.3·cos(−30°) = 0.3464 + 0.2598 = 0.6062 m.
2. Height: z = 0.5 + 0.4·sin 30° + 0.3·sin(−30°) = 0.5 + 0.2 − 0.15 = 0.55 m.
3. Rotate the plane by θ₁: x = 0.6062·cos 60° = 0.3031 m, y = 0.6062·sin 60° = 0.5250 m.
4. Check with the DH product A₁·A₂·A₃ (α₁ = 90°): fourth column = (0.3031, 0.5250, 0.5500). ✓

Answer: (0.303, 0.525, 0.550) m.

Example 3 (error sensitivity). The tool of an arm is 1.2 m from the base axis. The base encoder has an error of 0.01°. Position error ≈ r·δθ = 1.2 × (0.01 × π/180) = 2.09 × 10⁻⁴ m ≈ 0.21 mm.

Common mistakes

• Using θ₂ alone instead of the cumulative angle θ₁ + θ₂ for the second link.
• Calculator in radian mode with angles in degrees (or the reverse).
• Multiplying link transforms in the wrong order — always base to tool, A₁·A₂·…
• Forgetting the tool transform, so the computed point is the flange, not the gripper tip.
• Thinking FK can have several answers — it is unique; multiplicity belongs to IK.
• Using r·δθ with δθ in degrees.

For GATE ME

Most questions are planar 2R or 3R arms: find x, y or the tool angle for given joint angles, or the distance of the tool from the base. Occasionally a DH table is given for a 2- or 3-joint arm. Practise fast, careful trigonometry with cumulative angles and sanity-check with the reach limits |L₁ − L₂| ≤ r ≤ L₁ + L₂.

Quick check

1. Planar 2R, L₁ = 2 m, L₂ = 1 m, θ₁ = 0°, θ₂ = 90°. Tool position?
2. Is forward kinematics unique?
3. What is the tool orientation of a planar 3R arm with θ = 30°, 40°, −20°?
4. Order of multiplication for ⁰T₃?
5. A 0.001 rad error at a joint 0.8 m from the tool causes about how much position error?

Answers: 1. (2, 1) m; 2. Yes; 3. 50°; 4. A₁·A₂·A₃; 5. 0.8 mm.