Table of Contents

Configuration Space

Degrees of Freedom of a Rigid Body

In mechanical systems (and sometimes in electrical analogs), we often ask:
Where is the system located, and how is it oriented?

The answer is expressed in terms of degrees of freedom (DOFs): the minimum number of independent real-valued parameters required to specify the system’s configuration.

For robots, which we treat as assemblies of rigid bodies, these bodies are connected by joints (or links). Because the bodies are rigid, only a small number of parameters is needed to fully describe their configuration.

The collection of all possible configurations is called the configuration space (C-space), and its dimension is the number of degrees of freedom.

Translational vs. Angular DOFs in $N$ Dimensions

There are two common ways to count DOFs for a rigid body in $N$-dimensional Euclidean space:

Method 1 (classical counting):

• The first point requires $N$ parameters.
• The second point requires $N-1$ (it must be distinct but constrained by the first).
• Continue down to $1$, for a total of $$\frac{N(N+1)}{2}$$ DOFs.

Since $N$ of these are translational, the remaining

are angular DOFs.

Method 2 (pairwise intuition):

• A rigid body always has $N$ translational DOFs.
• Each angular DOF is defined by a plane spanned by two independent translation directions.
• Therefore, the number of angular DOFs is simply $$\binom{N}{2}.$$

Example: In 5D space, there are

angular DOFs, in addition to the 5 translational ones.

Grübler’s Formula

For general mechanisms, the Grübler-Kutzbach criterion provides the DOF count:

where:

• $m$ = number of DOFs of an unconstrained rigid body (6 in 3D space, 3 in planar motion).
• $N$ = total number of links (including the ground).
• $J$ = number of joints.
• $f_i$ = degrees of freedom allowed by the $i$-th joint.

Common Joint Types (3D Space)

• Revolute (R): 1 DOF
• Prismatic (P): 1 DOF
• Helical (H): 1 DOF
• Cylindrical (C): 2 DOF
• Universal (U): 2 DOF (two revolute joints with orthogonal axes)
• Spherical (S): 3 DOF

Important Notes

• Constraints between two planar rigid bodies reduce DOFs based on relative motion limitations.
• Discrete states (e.g., “heads” or “tails”) are not DOFs — DOFs must be continuous real parameters.
• $P$ is sometimes used to denote the number of controllable DOFs (those actuated by inputs).

Example: The Human Arm

The human arm (considering the hand as a rigid body) has 7 degrees of freedom:

1. Shoulder joint – 3 DOFs (spherical)

   • flexion/extension
   • abduction/adduction
   • internal/external rotation

2. Elbow joint – 1 DOF

   • flexion/extension

3. Forearm (radioulnar joint) – 1 DOF

   • pronation/supination

4. Wrist joint – 2 DOFs

   • flexion/extension
   • radial/ulnar deviation (abduction/adduction)

Total = 7 DOFs

Another way of thinking about it is by supposing you ‘fix’ your hand to a grounded surface, like a table. Since your arm can still move, it must have more than 6 degrees of freedom. In particular, it can only move in one way, so 6 + 1 is 7.

From Grubler’s formula, we have N = 5, m = 6, J = 7 => $6(5 - 1 - 7) + \sum_{i=1}^J f_i = 7$, so sum of freedoms provided by the joints is 25. In addition, constaints imposed by joints is $6\cdot4 - 7 = 17$.

Spaces and Representations

Charts and Atlases

Charts are local coordinate representations that handle singularities better than global representations. Singular points are locations where certain properties are not well-defined.

Multiple charts together form an atlas, providing complete coverage of the space. In general, we describe configuration spaces using more coordinates than strictly necessary, then subject them to appropriate constraints.

Types of Constraints

Holonomic vs. Nonholonomic Constraints

Holonomic Constraints: If there are $N$ coordinates and $k$ independent holonomic constraints, they reduce an $n$-dimensional configuration space to $(n-k)$ degrees of freedom.

Pfaffian Constraints: Constraints on velocity, such as $A(\theta)\dot{\theta} = 0$. These are nonholonomic if the velocity constraints do not restrict the configuration space itself.

Task Space vs. Workspace

Task Space: The space in which a task is most naturally represented, independent of the specific robot design.

Workspace: The specification of the reachable space for a particular robot configuration.

Representation Types

Implicit Representation: Uses more parameters than necessary (overcomplete) Explicit Representation: Uses minimal parameters for complete description

Example: Differential Drive Robot

Consider a differential drive robot with non-slip wheel constraints:

System Analysis

1. Controls: The robot has 2 control inputs - the left and right wheel speeds
2. Velocity Constraints: There are 3 Pfaffian constraints on the system velocities
3. Constraint Classification:
   • 1 nonholonomic constraint: No-slip rolling condition
   • 2 holonomic constraints: Wheel-ground contact conditions

This constraint structure determines the robot’s motion capabilities and planning requirements.