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.
Understanding Robot Kinematics
A useful way to understand an arm is to follow its coordinate frames. Each joint has its own local direction. When the first joint rotates, every link farther along rotates with it.
The second joint angle is measured from the first link, not usually from the room. This is why the angles are added as you move from the base toward the tool. Each link contributes a horizontal part and a vertical part.
Trigonometry finds those parts, then the parts are combined. The final tool direction matters as much as its location when a gripper must place a part, a pen must draw, or a welding torch must face a seam.
Inverse kinematics is harder because the same target can often be reached in different ways. A two or three link arm may have an elbow-up arrangement or an elbow-down arrangement. One arrangement may avoid a table while the other hits it.
Some targets are outside the workspace, so no joint settings can reach them. Other targets are reachable in position but not with the required tool direction. Real arms have joint limits too.
A shoulder may be unable to rotate through a full circle. Good robot programs check reachability and limits before commanding motion.
Some arm positions are especially difficult. Near a fully stretched or fully folded shape, small changes in tool position can require large changes in joint angles. This is called a singular configuration.
At such a point, the arm loses a useful direction of motion or becomes very sensitive to measurement errors. Engineers often choose paths that stay away from these positions.
Computers commonly solve inverse kinematics by making a guess, measuring the remaining error, then adjusting the angles repeatedly. This numerical method is useful for complex arms, but it can settle on an unwanted solution if the starting guess is poor.
For a differential drive robot, equal wheel speeds produce straight travel. If one wheel moves faster, the robot follows a circular arc toward the slower wheel. If the wheels turn at equal speeds in opposite directions, the robot can spin around a point between its wheels.
The wheel separation sets how sharply it turns for a given speed difference. Students meet this idea in classroom rover kits, robot vacuums, warehouse carts, and video game vehicles. Wheel encoders estimate distance by counting rotation, a process called odometry.
Odometry slowly drifts when wheels slip, tire sizes differ, or the floor is uneven. Sensors such as cameras, gyroscopes, and distance sensors help correct that drift. When learning kinematics, draw the reference axes, keep angle units consistent, state which direction counts as positive, and test calculations with simple cases such as zero joint angles or equal wheel speeds.
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.