Kinematic chains and pose

• Compute planar forward kinematics for a two-link chain, distinguish pose from force, and diagnose frame or angle-convention errors.

• vectors-frames
• mechanisms-kinematics

A kinematic chain connects rigid links with joints. Each joint permits a kind of motion: a revolute joint changes angle, while a prismatic joint changes extension. A desk lamp, printer carriage, and robot arm can all be described this way. Kinematics asks where parts are and how geometry relates motions, without yet asking what forces or torques create them.

For a planar two-revolute-link arm with lengths L1 and L2, base-frame endpoint position is x = L1 cos θ1 + L2 cos(θ1 + θ2), y = L1 sin θ1 + L2 sin(θ1 + θ2). The second angle is relative to the first link, so its world orientation is their sum. This is forward kinematics: joint values produce pose. Inverse kinematics asks for joint values that reach a desired pose and may have two solutions, one solution at a boundary, or none.

Frames keep statements honest. A camera may report a target in camera coordinates while the arm controller needs base coordinates. Homogeneous transforms combine rotation and translation so chained relationships can be multiplied in order. Order matters because rotations and translations generally do not commute. Before choosing a 120 mm + 80 mm linkage, a builder should compare the reachable annulus: maximum reach L1 + L2 = 200 mm and minimum radial reach |L1 - L2| = 40 mm, ignoring collisions and joint limits.

A common misconception is that reaching a point proves the pose is usable. Joint limits, self-collision, tool orientation, cable bend, and singular geometry may rule it out. Another failure is mixing degrees with radian-based library functions or treating θ2 as an absolute world angle. Draw axes and a zero pose, state angle sign and units, calculate a simple configuration, then compare the physical chain. Kinematics predicts geometry, not payload capability or motor torque.

Problem

A planar arm has L1 = 120 mm, L2 = 80 mm, θ1 = 30°, and relative θ2 = 60°. Find its endpoint.

Steps

1. The second link’s world angle is θ1 + θ2 = 90°.
2. x = 120 cos 30° + 80 cos 90° = 103.9 mm + 0 mm = 103.9 mm.
3. y = 120 sin 30° + 80 sin 90° = 60.0 mm + 80.0 mm = 140.0 mm.

Result

The endpoint is approximately (103.9 mm, 140.0 mm) in the defined base frame.

Sanity check

Its radial distance is about 174.3 mm, between the chain’s 40 mm minimum and 200 mm maximum possible reach.

For L1 = L2 = 100 mm, can the endpoint (250 mm, 0) be reached in the plane?

A. Yes, because each joint can rotate 360° B. No, because the requested radial distance exceeds L1 + L2 = 200 mm C. Yes, if both angles are expressed in radians D. No, because two-link arms can reach only circular paths

Your answer: ______________________________

Reasoning:

Goal

Build a paper or cardboard two-link chain and compare predicted endpoint coordinates with observed geometry.

Equipment

• Cardboard strips or two rulers
• two paper fasteners
• protractor
• graph paper
• ruler
• calculator

Procedure

1. Make 120 mm and 80 mm links with pivot holes centered at their ends; mark the base x- and y-axes.
2. Set θ1 = 30° and relative θ2 = 60°, then mark the endpoint without moving the paper.
3. Calculate the predicted point and measure the actual marked coordinates with an uncertainty based on ruler and joint looseness.
4. Introduce a deliberate model fault by treating θ2 as an absolute 60° angle; plot that prediction and use the mismatch to identify the convention error.
5. Try one point near 200 mm radius and note joint alignment and sensitivity to small angle changes.

What to observe

• Expected: the correct relative-angle prediction lies near (104 mm, 140 mm).
• Record actual coordinates and estimated reading/joint uncertainty; do not relabel the calculation as a measurement.
• Expected fault signature: the absolute-angle mistake predicts the second link at 60° rather than 90°, shifting both coordinates.

Transfer challenge

Choose two link lengths for a camera mount that must reach points 70–180 mm from its base, then list joint-limit and cable constraints that geometry alone omits.

Cautions

• Use blunt tools or supervised cutting for cardboard; this unpowered model validates geometry only, not structural strength.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: