Motion and Degrees of Freedom

Opening challenge: A rover can eventually reach any pose on a flat floor, so why can it not move directly sideways right now?

Degrees of Freedom Count Configuration

A robot's configuration is the smallest set of numbers needed to specify where all its rigid parts are. The set of possible configurations is configuration space, or C-space. Its dimension is the robot's number of degrees of freedom (DOF).

A free rigid body in a plane has three DOF: position $x$, position $y$, and orientation $\theta$. In three-dimensional space it has six: three translations and three rotations. Constraints remove independent choices. A hinge permits one rotation; a slider permits one translation.

Configuration Is Not Motion

Configuration $q$ answers where. Velocity $\dot q$ answers how fast and in which direction configuration is changing. Acceleration $\ddot q$ answers how velocity is changing. Units expose mistakes: position may be metres or radians, velocity metres per second or radians per second, and acceleration metres per second squared or radians per second squared.

For a planar rover:

$q=[x,y,\theta]^T$

It has three configuration DOF. A differential-drive rover normally commands forward speed $v$ and yaw rate $\omega$. With ideal no-side-slip wheels:

$\dot x=v\cos\theta,\qquad \dot y=v\sin\theta,\qquad \dot\theta=\omega$

At $\theta=0$, it cannot command an instantaneous world-$y$ slide. Yet by driving and turning, it can reach different $y$ positions later. Reachable eventually is not the same as movable directly now.

DOF Is Not Actuator Count

DOF counts independent configuration coordinates; actuators count independent control inputs in the chosen model. A system is underactuated when its available inputs cannot independently command every modeled DOF. The rover has three configuration DOF but two usual inputs, $v$ and $\omega$. Wheel constraints couple the motion.

Joint limits reduce range but do not normally change DOF inside that range. Locking a joint adds a constraint and can reduce DOF. Singular configurations can temporarily remove useful motion directions even when the mechanism's nominal DOF count is unchanged.

Worked Evidence Check

A rover starts at $(0,0,0)$ and executes $v=1\text{ m/s}$, $\omega=0$ for two seconds under an ideal constant-velocity model. It reaches $(2,0,0)$. If it instead commands $v=0$, $\omega=\pi/2\text{ rad/s}$ for one second, it remains at $(0,0)$ while yaw becomes $\pi/2$. Combining timed forward and turning segments produces sideways displacement without sideways slip.

Retrieval Pause

1. What does DOF count?
2. Why does a planar differential-drive rover have three configuration DOF but only two common velocity commands?
3. Explain why “cannot slide sideways now” does not mean “cannot reach a point to the side.”

Sources and Further Learning

• Lynch and Park, Degrees of Freedom of a Rigid Body, Northwestern Modern Robotics, retrieved 2026-08-19.
• Lynch and Park, Degrees of Freedom of a Robot, Northwestern Modern Robotics, retrieved 2026-08-19.
• Lynch and Park, Configuration and Velocity Constraints, Northwestern Modern Robotics, retrieved 2026-08-19.
• MIT, Underactuated Robotics: Fully Actuated vs Underactuated Systems, retrieved 2026-08-19.

Sources are linked and paraphrased. The teaching visual is an original HH asset.