KEY CONCEPTS & VOCABULARY

Trigonometry & Math Foundations

• Right Triangles & Trigonometric Ratios:
  Robotic arms (especially 2-joint linkages) are often modeled as a chain of right triangles. Understanding and applying sine, cosine, and tangent (SOH-CAH-TOA) is fundamental to solving robotic movement problems.

• Trigonometric Laws:
  The Law of Sines and the Law of Cosines are essential for finding unknown sides or angles in robot linkages (especially for non-right triangles).

• Coordinate Geometry:
  Trigonometry connects joint angles and arm segment lengths with positions on a Cartesian (x, y) grid, and vice versa.

• Angles & Units:
  All robotic movement is described in angles (degrees or radians). Know when to use each: degrees for sketches and communication, radians for programming and calculations.

• Graphing Trig Functions:
  Connect real robot motion to graphs of sine and cosine (visualizing amplitude, period, and phase shift).

Robotics & Kinematics

• Kinematics:

  • Forward Kinematics (FK): Find the tip position of a robot arm based on known joint angles and lengths.

  • Inverse Kinematics(IK): Find joint angles needed to place the tip at a target position.
    Both rely on trigonometry to relate angles and positions.

• Robot Arm Structure:
  Links, joints, pivots, and how they combine for movement.

• Degrees of Freedom:
  The number of independent movements a robot arm can make.

• Constraints of Motion:
  Range limits, precision, payload, and feasibility of robotic movement.

Engineering Design & Problem Solving

• Mathematical Precision:
  Trigonometry enables precise, repeatable robotic motion, which is critical for tasks requiring accuracy.

• Real-World Application:
  Trigonometry powers every movement: calculating lift, reach, angles, avoiding obstacles, and meeting user needs.

• Design Criteria & Constraints:
  Balancing user requirements with limits on materials, time, workspace, and mechanics.

• Problem Framing:
  Start by interviewing users and defining a real-world repetitive task the robot can address.

• Iteration & Optimization:
  Test prototypes, collect movement and math data, refine designs, and use trigonometric models to improve efficiency, accuracy, and usability.

• User-Centered Design:
  Always connect design decisions back to how the robot helps the real person.

Mathematical Modeling & Data

• Motion Sketching:
  Diagram tasks, arm movements, and ranges on coordinate grids or with triangles.

• Data Collection:
  Measure angles, distances, and timing, using protractors, video, or simple tools, to validate and tune models.

• Verification:
  Compare predicted motion (using math/models) to actual robot performance.

Project-Based Learning Practices

• Driving Question:
  "How can we use trigonometry to design a robotic arm that helps a real person with a repetitive task?"

• Need-to-Know Wall:
  Student-driven inquiry into which math, design, and engineering concepts need to be mastered.

• Collaboration & Peer Coaching:
  Working together to sketch, problem-solve, and give/receive structured feedback.

• Final Presentation:
  Clearly communicating your design solutions, mathematical reasoning, and how your work helps users.

Geometry & Triangles

• Right Triangle: A triangle with one angle equal to 90°.

• Oblique Triangle: A triangle that does not have a right angle.

• SOHCAHTOA: A mnemonic for the trigonometric ratios of sine (Opposite/Hypotenuse), cosine (Adjacent/Hypotenuse), and tangent (Opposite/Adjacent) in a right triangle.

• Inverse Trigonometry: Functions (sin-1, cos-1,tan-1) are used to find the angle when given two sides of a triangle.

• Law of Sines: A formula for solving oblique triangles when you know two angles and a side (AAS, ASA).

• Law of Cosines: A formula for solving oblique triangles when you know three sides (SSS) or two sides and the included angle (SAS).

• Sine (sin): In a right triangle, opposite side ÷ hypotenuse.

• Cosine (cos): In a right triangle, adjacent side ÷ hypotenuse.

• Tangent (tan):  In a right triangle, opposite side ÷ adjacent side.

• Hypotenuse: The side opposite the right angle in a right triangle.

• Adjacent Side: Side next to the angle of interest.

• Opposite Side: Side across from the angle of interest.

The Unit Circle & Angles

• Unit Circle: A circle with a radius of 1, centered at the origin, used to model circular motion and define trigonometric functions.

• Degrees: A unit for measuring angles in which a full circle is 360°.

• Radians: A unit for measuring angles in which a full circle is 2π radians. It relates an angle to the ratio of a circle's arc length to its radius.

• Arc Length: The distance along the curved edge of a circle's circumference.

Graphing & Modeling

• Periodic Function: A function that repeats its values in regular intervals or periods.

• Amplitude (A): The maximum distance a periodic function moves from its midline. In a robot arm, it represents the range of motion.

• Period (P): The time or angle required for one complete cycle of a periodic function. In a robot arm, it represents the speed of the motion.

• Midline (D): The central horizontal axis of a periodic function. In a robot arm, it represents the arm's resting position.

• Phase Shift (C): A horizontal translation of a periodic function. In a robot arm, it represents a delay or an early start in the motion.

• Vertical Shift (D): A vertical translation of a periodic function. In a robot arm, it represents the arm's resting position.

• Trigonometric Identity: An equation involving trigonometric functions that is true for all values of the variables for which the functions are defined.

Trigonometry & Math Terms

• Angle: The space (measured in degrees or radians) between two intersecting lines or surfaces.

• Radians: A unit of angle commonly used in trigonometry (1 radian ≈ 57.3°).

• Forward Kinematics: Using joint angles and link lengths to calculate the position of the end of the robot arm.

• Inverse Kinematics: Calculating what joint angles are needed to place the end of the robot arm at a specific location.

• Law of Sines/Cosines: Equations relating the angles and sides of triangles—used to solve for unknown sides/angles in robot arms.

• Coordinate Grid/System: Using (x, y) or (x, y, z) positions to locate points in space.

Robotics & Engineering Terms

• Actuator: A device, like a motor, that creates movement in a robot.

• Arm: The interconnected links and joints that make up a robotic manipulator (the “arm” that moves the end tool).

• Axis: A direction of movement (robots can move or rotate along several axes).

• Base: The foundation or platform to which a robot arm is connected.

• Compliance: The ability of a robot/part to flex in response to a force, rather than being completely rigid.

• Degrees of Freedom (DoF): The number of independent ways a robot arm can move (for example, rotate, extend, lift).

• End-Effector: The tool or gripper at the end of a robot arm that performs a task.

• Gripper: A type of end effector designed to grasp or hold objects.

• Joint: A pivot or hinge that allows parts of a robot arm to move (can be rotational or linear).

• Kinematics: The study of how things move, including linkages, angles, and positions in robot arms, without considering the forces or torques that cause the motion.

• Forward Kinematics (FK): Calculating the position of the arm's tip from its known joint angles.

• Inverse Kinematics (IK): Calculating the joint angles needed for the arm to reach a desired target position.

• Link: A solid segment between two joints of a robot arm.

• Linkage: A set of rigid links connected by joints, used in mechanisms like robot arms.

• Manipulator: The entire robot arm, including joints, links, and the end effector.

• Payload: The maximum weight a robot arm can carry without losing accuracy.

• Periodic Motion: Motion that repeats in a regular cycle.

• Range of Motion: How far a joint can rotate or extend.

• SCARA: Selective Compliance Articulated Robot Arm—a robot with a specific joint and movement pattern.

• Sensor: A device that detects or measures events or changes in the environment (e.g., light, touch, force, color).

• Servo (Servomechanism): A motor with feedback that keeps the robot’s position or speed under precise control.

• Stepper Motor: A motor that moves in precise steps, often used in robotic arms.

• Trajectory: The path that a point (like a robotic arm tip) follows through space.

Design & Coding Terms

• Constraint: A limit or restriction placed on the solution (e.g., size, budget, available materials).

• Conditional: A coding concept where an action occurs only if a specific condition is met (if-then logic).

• Criterion: A principle or standard by which something is judged or evaluated. The criteria are the goals the design must meet to be successful.

• Decompose: Breaking down a complex problem into smaller, more manageable parts.

• Design Criteria: The goals or features that a project must meet (e.g., cost, safety, speed).

• Engineering Design Process (EDP): A systematic approach to problem-solving that guides engineers from identifying a problem to designing, prototyping, and testing a solution.

• Empathy: The ability to understand and share the feelings of another person. In this project, it means understanding the user's needs and challenges to design a better solution.

• Feedback: Input from users, peers, or tests used to improve a design.

• Iteration: Repeating steps to refine and improve a design or solution.

• Optimization:  Making a solution as effective or efficient as possible.

• Problem Statement: A clear description of the challenge to solve.

• Prototype: A first or rough version of a design, built to test ideas.

• Pseudocode: A mix of programming logic and everyday language used to outline code and processes, a high-level description of a computer program's logic.

• Testing:  Evaluating how well a solution works.

• User Needs: The specific requirements or problems of the intended user.

Project Vocabulary

• Driving Question: The big, guiding question of the project.

• Need-to-Know (N2K) Questions: Student-generated questions that guide inquiry.

• Milestone: A key checkpoint in the project timeline.

• Final Presentation: A structured way of communicating the project solution.