Robotics uses math to turn sensor readings, motor commands, and physical measurements into controlled motion. This cheat sheet covers the math and trigonometry students need when building, programming, and testing robots. It is useful for drive trains, arms, odometry, navigation, and basic mechanism design.
The goal is to give quick access to formulas that connect code to real robot behavior.
The most important ideas are angle measurement, right-triangle trigonometry, vectors, coordinate systems, and motion relationships. Students should know how to switch between degrees and radians, use sin, cos, and tan, and break motion into x and y components. Robotics also depends on wheel circumference, gear ratios, speed, torque, and rotation formulas.
These tools help predict how far a robot moves, how fast it turns, and where it is on the field.
Key Facts
- Convert degrees to radians with radians = degrees x pi / 180.
- Convert radians to degrees with degrees = radians x 180 / pi.
- For a right triangle, sin(theta) = opposite / hypotenuse, cos(theta) = adjacent / hypotenuse, and tan(theta) = opposite / adjacent.
- The distance between two points is d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Vector components from a magnitude and direction are x = r cos(theta) and y = r sin(theta).
- Wheel circumference is C = pi d, and travel distance = wheel rotations x pi x wheel diameter.
- For gears or pulleys, output speed = input speed x input teeth / output teeth, while output torque = input torque x output teeth / input teeth.
- Average speed is v = distance / time, and angular speed is omega = angle / time when angle is measured in radians.
Vocabulary
- Radian
- A unit of angle measure based on arc length, where one full circle equals 2 pi radians.
- Vector
- A quantity with both magnitude and direction, such as robot velocity or displacement.
- Coordinate Plane
- A grid that uses x and y values to describe position on a flat surface.
- Odometry
- A method of estimating robot position by tracking wheel movement and rotation over time.
- Gear Ratio
- A comparison of gear sizes or tooth counts that changes speed and torque between input and output.
- Angular Velocity
- The rate at which an object rotates, usually measured in radians per second or degrees per second.
Common Mistakes to Avoid
- Using degrees when a formula or programming function expects radians is wrong because trig functions in many languages use radians by default.
- Forgetting to use wheel diameter in the circumference formula is wrong because C = pi d, not pi r, unless you first convert radius to diameter.
- Mixing units such as inches, centimeters, seconds, and minutes in one calculation is wrong because formulas only work when units are consistent.
- Swapping sine and cosine for vector components is wrong because x = r cos(theta) and y = r sin(theta) when theta is measured from the positive x-axis.
- Assuming a larger gear always makes a robot faster is wrong because a larger output gear usually increases torque while reducing output speed.
Practice Questions
- 1 Convert 135 degrees to radians.
- 2 A robot wheel has a diameter of 10 cm and turns 8 rotations. How far does the robot travel, ignoring wheel slip?
- 3 A robot moves 5 m at an angle of 30 degrees above the positive x-axis. Find the x and y components of its displacement.
- 4 A robot's calculated odometry position does not match its real position after driving across the field. Explain two physical reasons this could happen.