Why use CCD instead of an analytic formula?

A 3R spatial arm with joint limits is awkward for a single closed-form solution. CCD is simple to implement: each joint rotates to pull the end effector toward the target, iterating until the error is small or progress stalls.

Yes. As a local method it can stall at joint limits, fold into awkward local minima, or slow near singularities. Re-seeding posture or relaxing limits may be needed — behavior you can provoke by dragging the target aggressively.

What do the helix and figure-8 paths test?

They force continuous tracking through changing orientations and workspace heights, revealing lag, folding of links, and whether the iterative solver keeps up as the target moves.

3-Link 3D Arm Inverse Kinematics (CCD)

Continuation of two-link-arm-ik into 3D: 3 revolute joints (yaw + 2 pitches) solved with constrained Cyclic Coordinate Descent. Drag target in 3D or follow a helix / lemniscate / figure-8 trajectory.

About this model

A spatial 3R arm with yaw and two pitch joints reaches a 3D target by inverse kinematics. This simulator extends planar two-link IK into three dimensions and solves joint angles with constrained Cyclic Coordinate Descent (CCD): iteratively adjusting one joint at a time to reduce end-effector error while respecting joint limits. Drag the target in 3D or command helix, lemniscate, or figure-8 trajectories and watch the arm track. CCD is a local geometric heuristic — not an analytic closed-form IK, not Jacobian pseudoinverse control, and not collision-aware motion planning. Move the target through reachable and near-singular postures to see convergence behavior and joint-limit interactions.

Who it's for: Robotics, mechatronics, and advanced kinematics courses on iterative inverse kinematics.

Key terms

Inverse kinematics
Cyclic Coordinate Descent
3R manipulator
Joint limits
End effector
Trajectory tracking

How it works

3-link 3D robot arm with 3 revolute joints — base yaw (about Z), shoulder pitch and elbow pitch — solved with Cyclic Coordinate Descent (CCD) inverse kinematics. CCD walks from the end-effector back to the base, rotating each joint by the angle that would best align its remaining chain with the target — projected onto the joint's allowed hinge axis to honor the kinematic constraints. Drag the pink target around in 3D, run an automatic helix / lemniscate / figure-8 trajectory, and watch the joint angles tracked live. Continuation of `engineering/two-link-arm-ik` into 3D.

Key equations

CCD per joint j (axis a): align proj_a(e−p_j) with proj_a(t−p_j)

δq_j = atan2(∥a×·∥, a··), q_j ← clamp(q_j + step · δq_j, q_lim)

Frequently asked questions

Why use CCD instead of an analytic formula?: A 3R spatial arm with joint limits is awkward for a single closed-form solution. CCD is simple to implement: each joint rotates to pull the end effector toward the target, iterating until the error is small or progress stalls.
Can CCD get stuck?: Yes. As a local method it can stall at joint limits, fold into awkward local minima, or slow near singularities. Re-seeding posture or relaxing limits may be needed — behavior you can provoke by dragging the target aggressively.
What do the helix and figure-8 paths test?: They force continuous tracking through changing orientations and workspace heights, revealing lag, folding of links, and whether the iterative solver keeps up as the target moves.

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3-Link 3D Arm Inverse Kinematics (CCD)

About this model

How it works

Key equations

Frequently asked questions

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About this model

How it works

Key equations

Frequently asked questions