Robot Configuration and Degrees of Freedom
Before analyzing the motion of rigid bodies, it’s important to clarify two key concepts:
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Robot Configuration: This refers to the complete specification of a robot’s position, typically described by the values of its joints and the arrangement of its links. The configuration defines the pose (position and orientation) of every part of the robot in space.
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Degrees of Freedom (DOF): The number of independent variables (such as joint angles or displacements) required to uniquely determine the robot’s configuration. In other words, DOF tells us how many independent ways the robot can move.
A robot’s configuration captures both its translation (movement in space) and rotation (orientation). For example, a robotic arm in 3D space may require several joint values to fully describe its pose.
The next section explores how to determine the degrees of freedom for rigid bodies, using simple examples to build intuition.
Consider a coin, and select three non-collinear points on the coin: \(A\), \(B\), and \(C\). When an \(x\)-\(y\) coordinate system is defined, the coin initially appears to have 6 degrees of freedom (each point has 2 coordinates). However, the distances between these points are fixed due to the rigidity of the body:
\[\begin{align*} d_{AB} &= \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \\ d_{AC} &= \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} \\ d_{BC} &= \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} \end{align*}\]- Once point \(A\) is fixed, point \(B\) must satisfy the distance constraint \(d_{AB}\), reducing the degrees of freedom. The position of \(B\) can be described by the angle \(\theta_{AB}\) it makes with the x-axis, i.e., on the circle centered at \(A\) with radius \(d_{AB}\).
- With both \(A\) and \(B\) fixed, point \(C\) must satisfy two independent distance constraints (\(d_{AC}\) and \(d_{BC}\)). This restricts \(C\) to at most two possible positions, but for a rigid body, its position is uniquely determined.
Therefore, the coin has 3 degrees of freedom, which can be specified by:
\[(x_A, y_A, \theta_{AB})\]where:
- \((x_A, y_A)\): the position of point \(A\) (can represent the center or a reference point on the coin)
- \(\theta_{AB}\): the orientation of the coin, defined as the angle between the x-axis and a reference line on the coin
Let’s extend this example to 3D space. A rigid body in 3D space has 6 degrees of freedom:
In 3D, the position of a point is defined by three coordinates \((x, y, z)\), or as a position vector:
\[\vec{r} = x \vec{i} + y \vec{j} + z \vec{k}\]The elevation \((\psi)\) is measured from the \(x\text{-}y\) plane to \(z\), and the azimuth \((\theta)\) is measured in the \(x\text{-}y\) plane.
- Point \(A\) has 3 degrees of freedom \((x, y, z)\).
- Point \(B\), being at a distance \(d_{AB}\) from \(A\), can be anywhere on the surface of a sphere centered at \(A\). Elevation and azimuth angles can be used to describe the orientation of point \(B\).
- Point \(C\) must satisfy two distance constraints (\(d_{AC}\) and \(d_{BC}\)). The intersection of two spheres is a circle, so \(C\) has 1 degree of freedom along this circle.
Therefore, the rigid body in 3D has 6 degrees of freedom, which can be specified by:
\[(x_A, y_A, z_A, \psi_{AB}, \theta_{AB}, \psi_{AC})\]The general rule to determine the degrees of freedom is:
\[\text{DOF} = \text{sum of freedoms of points} - \text{number of independent constraints}\]Example: Degrees of Freedom of a Hinged Door
Consider a door attached to a wall by a hinge. Recall that a rigid body in 3D space has 6 degrees of freedom (DOF): 3 for translation and 3 for rotation.
When a door is hinged to a wall:
- The hinge prevents translation in the \(x\), \(y\), and \(z\) directions.
- The hinge also restricts rotation about two axes (those perpendicular to the hinge axis).
- The only allowed motion is rotation about the hinge axis.
This means the hinge imposes 5 independent constraints (3 translational and 2 rotational), leaving the door with 1 degree of freedom: the angle of rotation \(\theta\) about the hinge.
In summary:
- Degrees of freedom: 1
- Allowed motion: Rotation about the hinge axis (\(\theta\))
This example illustrates how constraints reduce the degrees of freedom of a rigid body.
As a general rule, the grubler’s formula can be used to determine the degrees of freedom of a system of rigid bodies:
\[\text{DOF} = m \cdot (N - 1) - \sum_{i=1}^{J} c_i\]where:
- \(m\): The number of degrees of freedom of a rigid body in planar or spatial space
- \(N\): The links in the system
- \(c_i\): The number of constraints imposed by the joints
- \[f_i + c_i = m\]
- \(f_i\): The number of degrees of freedom offered by the joint
- \(J\): The number of joints in the system
Robot joints Configuration space: Topology and representation