Robotics and Basic Kinematics

Robotics combines mechanics, sensing, control, and software. Kinematics is the part that relates joint motion to tool or end-effector motion without first considering force.

Learning Objectives

By the end of this lesson, you should be able to define links, joints, degrees of freedom, coordinate frames, forward kinematics, inverse kinematics, workspace, and singularity.

Links, Joints, and Degrees of Freedom

A robot is built from rigid links connected by joints. A revolute joint rotates; a prismatic joint translates. Each independent joint variable adds a degree of freedom.

A planar two-link arm has two revolute joints and can place its tip in a 2D region, but not every point or orientation is reachable.

Forward Kinematics

Forward kinematics computes tool position from joint values. For a planar two-link arm:

$$
x=L_1\cos\theta_1+L_2\cos(\theta_1+\theta_2)
$$

$$
y=L_1\sin\theta_1+L_2\sin(\theta_1+\theta_2)
$$

If L1 = 0.25 m, L2 = 0.20 m, theta1 = 30 deg, and theta2 = 45 deg, the equations give the approximate tip location in the base frame.

Inverse Kinematics

Inverse kinematics works backward: find joint values that place the tool at a desired location. IK may have multiple solutions, no solution, or solutions that violate joint limits.

flowchart LR JOINTS["Joint angles"] --> FK["Forward kinematics"] --> POSE["Tool pose"] TARGET["Target pose"] --> IK["Inverse kinematics"] --> OPTIONS["Possible joint angles"]

Workspace and Singularities

Workspace is the region a robot can reach. Singularities are configurations where the robot loses useful motion direction or requires very high joint speed for small tool motion. Straight-arm configurations are common singularity risks in simple arms.

Motion Planning Basics

A robot should not simply jump from one pose to another. Motion planning respects joint limits, velocity, acceleration, collisions, cable routing, payload, and safe stop behavior.

Common Mistakes

Confusing motor angle with tool position.
Ignoring joint limits in inverse kinematics.
Planning through a singularity.
Moving without acceleration limits.
Forgetting the tool frame offset.

Summary

Kinematics is the map between joint space and task space. Forward kinematics predicts where the robot is; inverse kinematics finds joint commands for a target. Real robots also need limits, planning, calibration, and safety.

Mind Map

mindmap root((Robot Kinematics)) Core concept Joints create motion Frames describe pose Kinematics maps spaces Formulas x equals L1 cos theta1 plus L2 cos theta1 plus theta2 y equals L1 sin theta1 plus L2 sin theta1 plus theta2 DOF counts independent motion Applications Robot arms Gantries Mobile manipulators Pick and place Design rules Respect joint limits Avoid singularities Calibrate frames Plan acceleration Practical checks Reach target Tool offset Collision path Repeatability Common mistakes No frame convention IK outside workspace Ignoring limits Jump commands