Robotics and Basic Kinematics
Robotics combines mechanics, sensing, control, power electronics, software, and safety. Kinematics is the part that relates joint motion to tool or end-effector motion without first calculating forces. It answers questions such as "Where is the gripper if these joints move?" and "Which joint angles can reach this target?"
Kinematics is essential before control tuning. A motor angle is not automatically a tool position. Gear ratios, link lengths, coordinate frames, joint limits, backlash, calibration, and tool offsets all affect where the robot actually moves.
Learning Objectives
By the end of this lesson, you should be able to:
- Define links, joints, degrees of freedom, coordinate frames, tool pose, and workspace.
- Distinguish forward kinematics from inverse kinematics.
- Use basic planar two-link arm equations.
- Recognize multiple IK solutions, unreachable targets, and singularity risks.
- List motion-planning and safety checks before commanding a robot.
Links, Joints, and Degrees of Freedom
A robot is built from rigid links connected by joints:
- A revolute joint rotates and is usually measured in degrees or radians.
- A prismatic joint translates and is usually measured in meters or millimeters.
- A link is the mechanical body between joints.
- A degree of freedom is one independent motion variable.
A gantry with X, Y, and Z slides has three prismatic degrees of freedom. A simple planar arm with shoulder and elbow revolute joints has two degrees of freedom. A typical industrial arm often has six revolute joints so it can control 3D position and orientation.
Coordinate Frames
A coordinate frame defines an origin and axes. Common frames include:
| Frame | Meaning |
|---|---|
| Base frame | fixed to the robot base or machine |
| Joint frame | fixed to a moving joint or link |
| Tool frame | fixed to the gripper, cutter, nozzle, or camera |
| Work frame | fixed to a part, fixture, or table |
Frame mistakes are common. If the tool offset is wrong, the robot may report that the wrist is in the correct location while the actual gripper is offset by several centimeters.
Forward Kinematics
Forward kinematics computes tool position from joint values. For a planar two-link arm:
$$
x=L_1\cos\theta_1+L_2\cos(\theta_1+\theta_2)
$$
$$
y=L_1\sin\theta_1+L_2\sin(\theta_1+\theta_2)
$$
where:
L1andL2are link lengths in meters.theta1is the shoulder angle.theta2is the elbow angle relative to link 1.xandyare tool coordinates in the base frame.
Worked Example
Let L1 = 0.25 m, L2 = 0.20 m, theta1 = 30 deg, and theta2 = 45 deg.
$$
x=0.25\cos30^\circ+0.20\cos75^\circ
$$
$$
y=0.25\sin30^\circ+0.20\sin75^\circ
$$
Using cos30 = 0.866, cos75 = 0.259, sin30 = 0.5, and sin75 = 0.966:
$$
x\approx0.25(0.866)+0.20(0.259)=0.268\ \text{m}
$$
$$
y\approx0.25(0.5)+0.20(0.966)=0.318\ \text{m}
$$
This result is only as good as the link measurements, angle zero references, gearbox calibration, and frame convention.
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.
For a two-link arm, many reachable points have an "elbow up" and an "elbow down" solution. A real controller must choose one based on joint limits, obstacles, cable routing, payload, and current robot posture.
Workspace and Singularities
Workspace is the region a robot can reach. For a planar two-link arm, the approximate radial reach is between:
$$
|L_1-L_2| \le r \le L_1+L_2
$$
where r is distance from the base to the target. With L1 = 0.25 m and L2 = 0.20 m, the arm can reach radii from about 0.05 m to 0.45 m, before considering joint limits and physical collisions.
Singularities are configurations where the robot loses useful motion direction or requires very high joint speed for small tool motion. A fully stretched planar arm is a common singularity risk: motion perpendicular to the arm may be possible, but motion further outward is impossible and small target changes can demand large joint changes.
Motion Planning Basics
A robot should not simply jump from one pose to another. Motion planning must respect:
- joint limits and software travel limits;
- velocity, acceleration, and jerk limits;
- collision with fixtures, tools, cables, and humans;
- payload mass and center of gravity;
- singularity avoidance;
- safe stop and recovery behavior;
- calibration and homing accuracy.
For beginner systems, start with slow joint-space moves, verify each axis independently, and use conservative workspace limits before attempting coordinated motion.
Common Mistakes
- Confusing motor angle with tool position after a gearbox or belt.
- Mixing degrees and radians in trigonometric calculations.
- Ignoring joint limits in inverse kinematics.
- Planning through a singularity.
- Moving without acceleration limits.
- Forgetting the tool frame offset.
- Assuming a reachable point is safe to reach along any path.
- Calibrating zero positions visually instead of using repeatable references.
Summary
Kinematics is the map between joint space and task space. Forward kinematics predicts where the robot is from joint values; inverse kinematics finds joint commands for a target. Real robots also need calibrated frames, joint limits, motion planning, collision checks, singularity avoidance, and safety-rated behavior where humans or high energy are involved.
Further Reading
- Kevin Lynch and Frank Park, Modern Robotics.
- Peter Corke, Robotics, Vision and Control.
- ROS documentation, TF and robot state publisher concepts.
- ISO 10218 and ISO/TS 15066 for industrial and collaborative robot safety context.