Robot kinematics is the study of how a robot moves without focusing on the forces that cause the motion. It uses geometry, angles, positions, and velocities to predict where robot parts will be. This matters because a robot arm must know where its end effector is before it can weld, pick up an object, or draw a path.
For students, kinematics connects trigonometry and coordinate graphs to real engineering machines.
A 3-link planar robotic arm can be modeled as connected line segments that rotate at joints with angles θ1, θ2, and θ3. In forward kinematics, the joint angles and link lengths are known, and the goal is to calculate the end-effector position. In inverse kinematics, the desired end-effector position is known, and the goal is to solve for possible joint angles.
A differential drive robot uses left and right wheel speeds to control forward motion and turning, so its path also follows kinematic rules.
Key Facts
- Forward kinematics: given θ1, θ2, θ3 and link lengths L1, L2, L3, compute the end-effector position.
- Planar arm x-position: x = L1 cos(θ1) + L2 cos(θ1 + θ2) + L3 cos(θ1 + θ2 + θ3).
- Planar arm y-position: y = L1 sin(θ1) + L2 sin(θ1 + θ2) + L3 sin(θ1 + θ2 + θ3).
- End-effector orientation for a planar 3-link arm: φ = θ1 + θ2 + θ3.
- Inverse kinematics: given a desired position such as (x, y), solve for joint angles that can reach it, sometimes with more than one solution.
- Differential drive robot speed and turn rate: v = (vR + vL)/2 and ω = (vR - vL)/b, where b is the distance between wheels.
Vocabulary
- Kinematics
- Kinematics is the description of motion using position, velocity, angles, and time without analyzing forces.
- Forward Kinematics
- Forward kinematics calculates the position and orientation of a robot's end effector from known joint values.
- Inverse Kinematics
- Inverse kinematics finds joint values that place the robot's end effector at a desired position and orientation.
- End Effector
- An end effector is the tool or working part at the end of a robot arm, such as a gripper, welder, or pen.
- Differential Drive
- Differential drive is a robot drive system where two powered wheels move at different speeds to create straight or curved motion.
Common Mistakes to Avoid
- Using θ2 by itself for the second link direction is wrong because the second link angle is measured relative to the first link. Use θ1 + θ2 for link 2 in the global coordinate frame.
- Forgetting to convert degrees to radians is wrong when using many calculators or programming languages. Check the angle mode before evaluating sin and cos.
- Assuming inverse kinematics has only one answer is wrong because many robot arms can reach the same point with different joint configurations. Always consider elbow-up, elbow-down, or unreachable cases.
- Ignoring the wheelbase in differential drive turning is wrong because turn rate depends on the distance between the wheels. Use ω = (vR - vL)/b instead of only comparing vR and vL.
Practice Questions
- 1 A 2-link planar arm has L1 = 4 cm, L2 = 3 cm, θ1 = 30°, and θ2 = 60°. Calculate the end-effector position using x = L1 cos(θ1) + L2 cos(θ1 + θ2) and y = L1 sin(θ1) + L2 sin(θ1 + θ2).
- 2 A differential drive robot has vL = 0.20 m/s, vR = 0.50 m/s, and wheelbase b = 0.30 m. Calculate its forward speed v and angular velocity ω.
- 3 A 3-link planar arm has link lengths 5 cm, 4 cm, and 3 cm. Explain why a target point 20 cm away from the base cannot be reached, and describe what the maximum reach tells you.