Mechanical Engineering, Robotics & Workplace Automation

Frames, Transforms & Forward Kinematics

Coordinate frames and homogeneous transforms, composing them by their labels, forward kinematics for a planar two-link arm with a worked example, and treating every frame in a robot cell as a calibrated engineering interface.

  • 2 min
  • 4 steps
  • 3 questions
  • Lesson 55 of 78

In this lesson

  1. Pose as rotation plus translation
  2. Planar two-link arm
  3. Frames are engineering interfaces

Open alongside this lesson

Robots reason in coordinate frames. Stanford CS223A begins with spatial descriptions before forward kinematics, Jacobians, inverse kinematics, dynamics, and control 1.

Introduction to Robotics at MIT MIT's project model makes robotics an integration discipline: geometry, mechanisms, fabrication, sensing, controls, code, testing, and teamwork meet in one machine. Credit: MIT Department of Mechanical Engineering · All rights reserved; embedded from the official MIT upload · 2:50 · Source

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Pose as rotation plus translation

A homogeneous transform from frame B to frame A is

\[ {}^AT_B=\begin{bmatrix}{}^AR_B & {}^Ap_B\\0&1\end{bmatrix}. \]

It maps coordinates expressed in B into A. Composition follows the frame labels:

\[ {}^AT_C={}^AT_B{}^BT_C. \]

Matrix order matters because rotations generally do not commute. Write superscripts and subscripts until the chain cancels cleanly.

Base, shoulder, elbow, wrist, and tool coordinate frames are linked by homogeneous transforms to locate an end effector
Forward kinematics composes local joint-and-link relationships into one base-to-tool pose. Credit: StudyCorner original diagram · CC BY 4.0 · Source

For link lengths \(l_1,l_2\) and revolute joints \(q_1,q_2\), tool position is

\[ x=l_1\cos q_1+l_2\cos(q_1+q_2), \quad y=l_1\sin q_1+l_2\sin(q_1+q_2). \]

With \(l_1=0.40\) m, \(l_2=0.30\) m, \(q_1=30^\circ\), and \(q_2=45^\circ\), \(x\approx0.424\) m and \(y\approx0.490\) m. Check the result against maximum reach 0.70 m and a sketch of the quadrant.

Quick check

A two-link arm has l1 = 0.40 m, l2 = 0.30 m, q1 = 30°, q2 = 45°. About where is the tool?

Frames are engineering interfaces

Define base, world, fixture, part, camera, flange, tool, and task frames. Record who establishes each, how it is calibrated, when it can change, and what invalidates it. Many “robot accuracy” problems are actually loose fixtures, wrong tool-center-point data, or stale frame calibration.

Frame audit

Sketch a pick-and-place cell with every coordinate frame and transform. For each, label fixed by design, measured during setup, or estimated during operation. Then trace one point from camera pixels to part pose to robot base to tool command. Every unstated transform is a future integration defect.

Practice

What does a homogeneous transform encode?

Practice

What does forward kinematics compute?

Lesson complete

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Inverse Kinematics, Jacobians & Singularities

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Sources for this lesson
  1. 1
    CS223A / ME320 - Introduction to Robotics. Stanford University. verifiedCurrent physics-based syllabus covering spatial transformations, kinematics, Jacobians, dynamics, motion and force control, and vision-based control. Cited at: 2026 syllabus.

Further reading

  • Introduction to Robotics. MIT OpenCourseWare. verifiedMechanisms, kinematics, planning, dynamics, controls, actuators, sensors, networks, interfaces, embedded software, laboratories, and a team robot project.
  • Stanford Engineering Everywhere - CS223A Introduction to Robotics. Stanford University. verifiedFree lecture videos, transcripts, handouts, and assignments on robot kinematics, Jacobians, planning, dynamics, and control.
  • Introduction to Robotics at MIT. MIT Department of Mechanical Engineering. 2015. verifiedOfficial MIT MechE video showing the 2.12 project model and its integration of design, manufacturing, controls, programming, and teamwork.

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