Robot Coordinates and Planar Motion

Robotics and Automation · CHAPTER 1 / 5

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

Learning outcome: Represent a robot position using a defined coordinate frame.

Before you begin: Basic algebra, SI units and a willingness to sketch the problem.

1. Understand the idea

A position vector has meaning only relative to a frame. Define origin, axis directions and units before calculating motion. For a planar link rotating about a fixed joint, the endpoint lies on a circle. Degrees and radians must be handled consistently in software.

2. Write the model

x=L cosθ; y=L sinθ

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

For L=1 m and θ=30°, the endpoint is (0.866,0.500) m. At 60° it is (0.500,0.866) m. The distance from the origin remains 1 m.

4. Explore the model

Figure: calculated teaching cases; straight segments connect sampled points, not experimental observations. Read exact values below.

Illustrative scenario sweep — synthetic educational data
Angle (degrees)	x (m)	y (m)
0	1	0
30	0.866025	0.5
60	0.5	0.866025
90	0	1

Sweep angle and plot the endpoint. Check x²+y²=1 to catch unit or trigonometric errors.

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

Where is the endpoint at 180°?

Reveal the answer and check your work

(−1,0) m.

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

7. Common mistake

Many programming language trig functions expect radians.

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.