Forward Kinematics Calculator (2-Link Planar Arm)

What forward kinematics solves

A robot arm has joints (rotating motors, sometimes sliders) connecting rigid links. Forward kinematics is the easier of the two kinematics directions: given the joint angles, compute where the end-effector (the gripper tip or tool point) actually is. The reverse problem, inverse kinematics, is harder because multiple joint configurations can put the gripper at the same point.

For a 2-link planar arm rotating in a flat plane around the base, with link lengths L₁ and L₂ and joint angles θ₁ (shoulder) and θ₂ (elbow, measured relative to the upper arm), the end-effector position is:

x = L₁ cos(θ₁) + L₂ cos(θ₁ + θ₂) y = L₁ sin(θ₁) + L₂ sin(θ₁ + θ₂)

The end-effector orientation (the direction the gripper points, in the plane) is simply the sum of the two joint angles:

φ_end = θ₁ + θ₂

These three equations completely describe the position and pose of a 2-link arm. For a 3-link planar arm you would add a third term, and for a 6-DOF industrial arm you would chain together six 4×4 Denavit-Hartenberg transformation matrices.

Why a 2-link planar arm is the standard teaching example

Real industrial robots have 6 or 7 degrees of freedom and require Denavit-Hartenberg parameters to compute their kinematics. The math is the same in principle but the bookkeeping explodes. The 2-link planar case captures the essential ideas with the smallest possible complexity:

• Multiple links and joints
• Each link extends from the previous one
• Angles compound (θ₁ + θ₂ for the second link)
• Reachable workspace is a region of the plane (an annulus between |L₁ − L₂| and L₁ + L₂)
• Two solutions exist for most reachable points (elbow up vs elbow down configurations)

Once you understand the 2-link arm, the generalization to 6 DOF is mechanical (4×4 transformation matrices replace the 2×2 rotations) but conceptually identical.

The Denavit-Hartenberg parameters

For a multi-joint arm, each link has four DH parameters: (θ, d, a, α), where:

• θ = joint angle, the variable for revolute joints (or the only fixed parameter for prismatic joints)
• d = link offset (translation along the joint’s z-axis), the variable for prismatic joints
• a = link length (translation along the previous link’s x-axis after rotation)
• α = link twist (rotation about the previous link’s x-axis)

The 4×4 homogeneous transformation matrix for joint i:

T_i = [ cos(θ) −sin(θ)cos(α) sin(θ)sin(α) a·cos(θ) ] [ sin(θ) cos(θ)cos(α) −cos(θ)sin(α) a·sin(θ) ] [ 0 sin(α) cos(α) d ] [ 0 0 0 1 ]

The total transformation from base to end-effector is the product T = T₁ T₂ T₃ … T_n. The position of the end-effector in the base frame is the last column of T (rows 1 to 3). The orientation comes from the upper-left 3×3 sub-matrix.

For a 2-link planar arm with all α = 0 and all d = 0, the DH math collapses to the simple 2D formulas at the top.

Worked example: a SCARA-style 2-link arm

L₁ = 0.5 m, L₂ = 0.3 m. The arm starts straight at θ₁ = θ₂ = 0, with end-effector at (0.8, 0).

Rotate the shoulder to θ₁ = 90° (upward) keeping θ₂ = 0:

• x = 0.5 cos(90°) + 0.3 cos(90°) = 0 + 0 = 0
• y = 0.5 sin(90°) + 0.3 sin(90°) = 0.5 + 0.3 = 0.8
• End-effector is at (0, 0.8), pointing straight up.

Bend the elbow to θ₂ = −90° (forearm folds forward from the upper arm):

• x = 0.5 cos(90°) + 0.3 cos(90° + (−90°)) = 0 + 0.3 cos(0°) = 0 + 0.3 = 0.3
• y = 0.5 sin(90°) + 0.3 sin(0°) = 0.5 + 0 = 0.5
• End-effector is at (0.3, 0.5), gripper pointing horizontally (φ = 0°).

Each joint angle change moves the end-effector predictably in the plane.

Reachable workspace

A 2-link arm with link lengths L₁ and L₂ can reach any point within an annulus centered on the base. The outer boundary is when both links extend straight outward: radius = L₁ + L₂. The inner boundary is when the second link folds back on the first: radius = |L₁ − L₂|. Points outside this annulus are unreachable.

For L₁ = 0.5, L₂ = 0.3: outer reach 0.8 m, inner unreachable disk of 0.2 m radius. A point at distance 0.1 from base cannot be reached by this arm because the forearm cannot bend back tightly enough.

If L₁ = L₂ (equal-length links), the inner boundary collapses to zero and the arm can reach all the way to its base. This is sometimes called a “full-shoulder” design.

The two-solution problem

Most reachable points (except boundary points) have two valid joint configurations: elbow up and elbow down. Both put the gripper at the same (x, y), but the elbow joint is on opposite sides. This is the “inverse kinematics multi-solution” problem.

Forward kinematics does not have this ambiguity. Given specific joint angles, there is exactly one end-effector position. The ambiguity arises only when going backwards (from position to angles), which is why inverse kinematics is harder.

Where this is used

Industrial pick-and-place robots use forward kinematics to convert motor positions into Cartesian gripper coordinates, for collision checking and motion planning. Surgical robots compute the tool tip position from instrument-arm joint angles thousands of times per second to display the location to the surgeon.

In video game physics, character animation, and computer graphics, forward kinematics is what skeletal-animation rigs use: a hierarchy of bones rotated by user input, with each bone position computed as the parent’s rotation cascades down the chain. Inverse kinematics solvers in animation are slower and more iterative; forward is the cheap default.

In computer-aided design and 3D simulation, FK lets you preview where a manipulator will go before committing motor commands. It is the building block for inverse kinematics solvers (Jacobian methods iteratively use FK at slightly-shifted joint angles to compute partial derivatives).

Companion calculator

For the inverse problem (compute joint angles to reach a target Cartesian point), see the Inverse Kinematics Calculator. The two pair naturally: forward gives the workspace; inverse plans motion to specific targets.