Forward and Inverse Kinematics

Opening challenge: If a robot hand must reach one point, is there always exactly one set of joint angles that works?

Two Questions, Opposite Directions

Forward kinematics (FK) starts with joint configuration $q$ and computes an end-effector pose: $X=f_{kin}(q)$. Inverse kinematics (IK) starts with a desired end-effector pose and searches for a configuration $q$ that satisfies it.

For a planar two-link arm with link lengths $L_1,L_2$ and angles $\theta_1,\theta_2$:

$x=L_1\cos\theta_1+L_2\cos(\theta_1+\theta_2)$

$y=L_1\sin\theta_1+L_2\sin(\theta_1+\theta_2)$

With $L_1=L_2=1$, $\theta_1=0^\circ$, and $\theta_2=90^\circ$, FK gives $(x,y)=(1,1)$.

Why IK Is Not a Simple Reverse Button

The target $(1,1)$ can have an elbow-up and elbow-down solution. Some targets have one solution at a workspace boundary. Targets farther than $L_1+L_2$ are unreachable. A numerical IK solver can also fail even when a geometric solution exists because of its seed, timeout, or local minimum.

A pose solution is not automatically a safe motion. Useful IK must consider:

• joint limits and preferred posture;
• collisions and clearance;
• target position and orientation tolerances;
• continuity from the current configuration;
• solver timeout and failure handling.

Joint Space and Task Space

Joint space describes configurations such as $(\theta_1,\theta_2)$. Task space describes an outcome such as hand pose $(x,y,\phi)$. FK maps joint space to task space. IK searches from a task-space request back to joint space. Redundant arms may have infinitely many configurations that satisfy a lower-dimensional task, so an objective such as staying near a comfortable posture selects among them.

Misconception Check

• “IK always has one answer.” It may have many, one, or none.
• “If IK succeeds, the motion is safe.” IK returns a configuration, not necessarily a collision-free trajectory.
• “FK and IK are equally easy.” FK evaluation is normally direct; IK is a constrained search or optimization problem.

Retrieval Pause

1. What information flows into and out of FK?
2. Why can one target have elbow-up and elbow-down solutions?
3. What three checks should follow an IK solution before execution?

Sources and Further Learning

• MIT Robotic Manipulation, Basic Pick and Place: Forward Kinematics, retrieved 2026-08-19.
• MIT Robotic Manipulation, Motion Planning: Inverse Kinematics, retrieved 2026-08-19.
• MoveIt 2, moveit_kinematics, retrieved 2026-08-19.

Sources are linked and paraphrased. The visual is an original HH educational asset.