Robot Anatomy and Degrees of Freedom

1. Task Space vs Joint Space

A robotic manipulator lives in two distinct mathematical spaces simultaneously. Joint space describes the internal configuration of motor angles $q = (\theta_1, \theta_2)$. Task space (or operational space) describes the Cartesian position and orientation of the end-effector or payload: $(x, y) = (1.20\text{ m}, 0.60\text{ m})$. Actuators drive joint angles; tasks specify mug coordinates.

2. Links and Revolute Joints

The arm is composed of rigid structural bodies called links, connected by movable joints. In our standard two-link planar arm, the upper arm link has length $L_1 = 1.00\text{ m}$ and the forearm link has length $L_2 = 0.80\text{ m}$. Each revolute (1-DOF pin) joint permits rotation around a single axis perpendicular to the plane.

3. Counting Degrees of Freedom

Mobility measures the number of independent variables required to specify a mechanism's configuration. In a plane, an unconstrained rigid body has 3 degrees of freedom. Grübler's formula for a planar mechanism with $N$ links and $J$ single-DOF joints calculates:

$M = 3(N - 1 - J) + \sum_{i=1}^J f_i = 3(3 - 1 - 2) + (1 + 1) = 2\text{ DOF}$

Two revolute joints supply exactly 2 degrees of freedom to position the wrist in the plane.

4. The Reachable Workspace

The end-effector cannot visit arbitrary points in space. The geometry of the links restricts the reachable workspace to a circular annulus centered at the shoulder base. The maximum reach occurs when both links are fully extended:

$R_{\max} = L_1 + L_2 = 1.00 + 0.80 = 1.80\text{ m}$

The minimum reach occurs when the forearm folds completely back against the upper arm: $R_{\min} = |L_1 - L_2| = |1.00 - 0.80| = 0.20\text{ m}$. Targets outside $1.80\text{ m}$ or inside the $0.20\text{ m}$ inner hole have no kinematic solution.

5. Actuator Torques at the Joints

Holding a position requires motors to counteract gravitational moments. Torque is the cross product of lever arm distance and force: $\tau = r \times F$. The elbow motor generates torque $\tau_2$ to support the forearm and payload. The shoulder motor must generate a much larger torque $\tau_1$ because it carries the cumulative moment arm of both links and the mug.

6. Holonomic vs Nonholonomic Mobility

Degrees of freedom in configuration do not always equal instantaneous directions of movement. A planar arm is holonomic: its actuators can command instantaneous velocity in any valid joint direction. In contrast, a two-wheeled differential-drive mobile robot is nonholonomic: while its pose requires 3 coordinates $(x, y, \theta)$, rolling without slip forbids instantaneous sideways translation ($\dot{y}_r = 0$). It must turn before translating sideways.

7. Redundant Degrees of Freedom

When an arm has more degrees of freedom than the task demands (e.g. 3 planar joints for a 2D position target), the robot is kinematically redundant. The extra degree of freedom creates a mathematical null space: the arm can continuously reconfigure its internal elbow posture to dodge an obstacle without moving the grasped mug.

8. Mechanical Joint Limits

Real joints cannot rotate continuously through $360^\circ$. Internal wiring harnesses twist and shear if over-rotated, and structural casings collide. Physical rubber hardstops and software limit switches enforce bounded joint ranges, such as $-150^\circ \le \theta_2 \le +150^\circ$, carving out deadbands from theoretical workspaces.

9. Torque Limits Under Payload

Kinematic reachability does not guarantee dynamic feasibility. When the arm holds a $2.0\text{ kg}$ mug at maximum horizontal extension ($r = 1.80\text{ m}$), gravitational torque on the shoulder reaches $\tau = 1.80 \times (2.0 \times 9.81) \approx 35.3\text{ N}\cdot\text{m}$. If the shoulder actuator is rated for only $30.0\text{ N}\cdot\text{m}$, the motor saturates, overheats, and drops the load.

10. Where It Breaks

When the arm straightens completely at the workspace boundary ($\theta_2 = 0^\circ$), it enters a kinematic singularity. The links align, reducing the rank of the Jacobian matrix ($\det(J) = 0$). The manipulator completely loses its radial degree of freedom: no combination of joint velocities can produce outward Cartesian motion. Commanding outward velocity demands infinite joint acceleration.

The Quick Version

• Joint space specifies internal motor angles; task space specifies end-effector coordinates.
• Rigid links transmit forces between one-degree-of-freedom revolute pin joints.
• Grübler's formula counts independent mobility degrees from links and joint constraints.
• Link lengths establish an annular reachable workspace with minimum and maximum radii.
• Base actuators bear the cumulative gravitational moment of all subsequent links.
• Holonomic joints move instantaneously in any direction; nonholonomic bases cannot move sideways.
• Redundant joints provide a null space to alter posture without moving the hand.
• Mechanical stops and cabling harnesses restrict joint rotations to bounded angles.
• Heavy payloads at long lever arms exceed actuator motor torque ratings.
• Boundary singularities drop Jacobian rank and prevent motion in locked directions.