Mechanisms and planar kinematics

• Compute the endpoint position of a two-link planar chain from joint angles.
• Recognize mobility limits, singular alignments, and mechanical interference in a linkage.

• algebra-trig
• vectors-frames

A mechanism turns allowed motion at joints into useful motion elsewhere. Door hinges, printer carriages, windshield-wiper linkages, and camera gimbals constrain parts so the user does not have to guide every coordinate. Kinematics describes position, velocity, and acceleration without yet asking what forces produce them.

For a planar two-revolute-link chain with lengths L₁ and L₂ and joint angles θ₁ and θ₂, measured with θ₂ relative to link one, the endpoint is x = L₁ cos θ₁ + L₂ cos(θ₁ + θ₂) and y = L₁ sin θ₁ + L₂ sin(θ₁ + θ₂). Differentiating relates joint rates to endpoint velocity through a Jacobian: v = J(q)q̇. When the links align, the Jacobian loses rank in one direction. Near that singular configuration, a modest requested endpoint motion can demand very large joint rates or produce little leverage in a direction.

Mobility depends on links and joint constraints, but a count alone does not reveal collisions, joint limits, backlash, or assembly mode. A four-bar may have the nominal mobility to move yet lock because link lengths violate reach conditions or because a bolt head hits a neighbor. Prototype the motion envelope before choosing an actuator. For a desk linkage, moving the motor attachment inward reduces required travel but increases required force; geometry and torque must be considered together.

A common misconception is that reachable points form a uniformly controllable workspace. A two-link arm can reach radii between |L₁ - L₂| and L₁ + L₂ in the ideal plane, but joint limits and singularities remove useful regions. Another failure is mixing absolute and relative angles in the endpoint equations. Draw coordinate frames on each link, declare the angle convention, and test easy poses such as both angles zero before trusting code or CAD motion.

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 absolute angle is θ₁ + θ₂ = -30°.
2. x = 120 cos 30° + 80 cos(-30°) = 103.9 mm + 69.3 mm = 173.2 mm.
3. y = 120 sin 30° + 80 sin(-30°) = 60.0 mm - 40.0 mm = 20.0 mm.

Result

The endpoint is approximately (173.2 mm, 20.0 mm) in the base frame.

Sanity check

Its distance from the base is 174.4 mm, within the ideal radial reach of 40–200 mm.

For a two-link planar arm, both links become collinear and fully extended. Why is that pose special?

A. The arm gains an extra joint B. Small radial outward motion is impossible and the Jacobian is singular C. Motor torque becomes zero for every load D. The endpoint position becomes undefined

Your answer: ______________________________

Reasoning:

Goal

Map the real workspace and singular behavior of a cardboard two-link arm.

Equipment

• Cardboard strips
• Two paper fasteners or bolts
• Grid paper
• Protractor
• Pencil

Procedure

1. Build two measured links on a pinned base and define zero angle and positive rotation on the grid.
2. Predict endpoint coordinates for five joint-angle pairs, then mark actual pencil positions without forcing joints.
3. Move each joint by roughly 5° near a bent pose and near full extension; compare endpoint motion direction and distance.
4. Deliberate fault: add a cardboard stop that restricts one joint; diagnose why the measured workspace is smaller than the ideal annulus.

What to observe

• Expected: calculations and marks broadly agree, with greatest directional sensitivity changing by pose.
• Measure: actual link lengths, angles, endpoint coordinates, joint play, and stop locations; do not treat drawing error as theoretical reach.

Transfer challenge

Choose a safe camera pointing region that stays away from a linkage singularity and physical cable collision.

Cautions

• Round sharp cardboard corners and keep fingers out of closing linkage angles.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: