Arms, Levers, and Degrees of Freedom

An arm creates reach by moving links around joints. Every added joint creates more capability and more control complexity.

One joint changes torque with position

Consider a horizontal arm holding a load.

When the load is far from the pivot, gravity creates a large torque.

As the arm rotates toward vertical above the pivot, the gravitational torque from that load decreases.

The motor therefore experiences different load conditions at different arm angles.

A controller tuned in one position may behave differently in another.

Degrees of freedom

A two-joint planar arm can place its endpoint in many positions within a reachable region.

base ●── link 1 ──●── link 2 ── gripper
       joint 1      joint 2

Adding a wrist changes object orientation. Adding rotation at the base changes workspace again.

Capability grows, but so do:

• actuators;
• sensors;
• wiring;
• control states;
• collision possibilities;
• failure modes.

Workspace versus reach

Maximum reach is not the same as useful workspace.

A robot may technically reach a point but only through a collision, unstable pose, weak torque configuration, or impossible joint angle.

Mechanical limits are good

Hard stops and software limits can protect the robot from commanding dangerous geometry.

A design should define:

• allowed joint range;
• sensor reference;
• homing method if needed;
• behavior when the limit is reached.

Joint map

Design a two-degree-of-freedom arm for a simple tabletop task.

Draw three poses:

1. home;
2. pickup;
3. place.

For each, mark:

• joint angles approximately;
• payload position;
• likely high-torque joint;
• one collision or limit concern.

Do not optimize yet. First understand the geometry you are asking the robot to control.

Every joint adds capability and responsibility

A degree of freedom is an independent motion the mechanism can command.

A simple rotating arm has one major joint variable. A two-link planar arm may have two joint angles:

shoulder angle = θ1
elbow angle    = θ2

Many different pairs of angles can place the end effector in different positions, and some target positions may be unreachable.

More degrees of freedom increase flexibility. They also increase:

• actuators;
• sensors;
• wiring;
• control complexity;
• collision possibilities;
• calibration work;
• failure modes.

Torque changes with posture

An arm holding the same object can require very different joint torque depending on geometry.

A 20 N load held 0.10 m horizontally from a joint produces an ideal moment near:

τ = 20 N × 0.10 m = 2.0 N·m

At 0.40 m:

τ = 20 N × 0.40 m = 8.0 N·m

Same object. Four times the lever arm. Four times the ideal load torque.

That is why mechanism design cannot choose an actuator from payload mass alone. The worst relevant posture matters.