Kinematics

Kinematics is the geometry of motion: where the parts of a robot are, given how its joints are set, without worrying about forces. It’s the bridge between “motor angles” and “position in the world.”

Drag the two joint angles and watch the arm’s tip (end-effector) move. The dashed circle is everything it can reach.

θ₁ (shoulder) = 50°θ₂ (elbow) = 35°

end-effector x = 59y = 131x = L₁cos θ₁ + L₂cos(θ₁+θ₂)

Forward kinematics

Forward kinematics (FK) computes the end-effector position from the joint angles — a direct calculation. For this two-link arm it’s just trigonometry (see the Math track):

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

Each joint adds a rotation; you chain them along the links. Real robots stack many such transforms (often as matrices) down the kinematic chain.

Inverse kinematics

The harder, more useful direction: inverse kinematics (IK) asks “what joint angles put the tip at this target?” Unlike FK, IK can have no solution (target out of reach), one, or many (the elbow can often go “up” or “down” to the same point). It’s solved with geometry or iterative numerical methods, and it’s how you tell an arm “go pick up that cup.”

Degrees of freedom

Each independent joint is a degree of freedom (DOF). More DOF means more ways to reach a pose (useful for avoiding obstacles) but a harder IK problem. A human arm has 7 DOF; this toy has 2.

Takeaways

• Forward kinematics: joint angles → end-effector position (direct, via trig).
• Inverse kinematics: target position → joint angles (may have 0, 1, or many solutions).
• Degrees of freedom set both flexibility and the difficulty of control.