Two-Link Forward Kinematics

Robotics and Automation · CHAPTER 2 / 5

Level: introductory undergraduate · Study session: 15–20 minutes including practice.

Learning outcome: Calculate a planar arm endpoint from joint angles.

Before you begin: Complete earlier chapters in this discipline; revise algebra and units as needed.

1. Understand the idea

For a two-link serial arm, the second link orientation is measured relative to the first joint frame. Add the first-link displacement to the second-link displacement expressed in the base frame. Forward kinematics maps joint angles to position; inverse kinematics solves the opposite problem and may have multiple solutions.

2. Write the model

x=L1cosθ1+L2cos(θ1+θ2); y=L1sinθ1+L2sin(θ1+θ2)

Define the symbols from the worked case before substituting values. Keep units consistent, record the assumptions and round only after the calculation.

3. Follow a worked example

With L1=L2=1 m, θ1=0° and θ2=90°, the endpoint is (1,1) m. Using θ2 alone as the global second-link angle works here only because θ1 is zero.

4. Explore the model

Concept diagram: specify the case, evaluate its model, then verify against the scenario table.

Illustrative scenario sweep — synthetic educational data
θ1 (degrees)	θ2 (degrees)	Endpoint (m)
0	0	(2,0)
0	90	(1,1)
90	0	(0,2)

Keep θ1=0° and sweep θ2. Draw each link separately before joining the endpoints.

5. Reproduce it yourself

Copy the input cases into Excel or a notebook and name each quantity and unit.
Recalculate each result from the stated equation or rule. For qualitative cases, explain the state or partition decision instead.
Change one input, predict the direction of change, then calculate again.
Write one sentence explaining where the model stops being valid.

6. Check your understanding

What is the endpoint at θ1=0°, θ2=180°?

Reveal the answer and check your work

(0,0) m in the ideal zero-thickness model.

Compare your result, units and reasoning. If they differ, return to the equation and test the worked case first.

7. Common mistake

Geometric reachability does not ensure collision-free motion.

Further study and academic note

Explore the discipline reference and further learning resources. This is an original IIMEEE introductory teaching note with synthetic examples; it is not a peer-reviewed research result or a professional design specification. Figures were constructed from the listed teaching cases. Resource links support further study and do not imply endorsement.