Kinematics

Kinematics

• Kinematics is very important for modeling robot position, frames, links, and joints.

• Kinematics involves:

  • Describing a task.

  • Using models to determine the position and orientation of the end effector.

  • Transforming descriptions between links.

  • Representing position and orientation.

Manipulators

• A manipulator is defined by a set of links connected through joints.

  • The first link is fixed (the base).

  • The last link is the gripper (end effector).

• The purpose of the manipulator is to move the gripper in space for manipulation, though links themselves can be used.

Joints

• Two primary types of joints:

  • Revolute joints: rotation about a fixed axis (one degree of freedom).

  • Prismatic joints: translation along a fixed axis (one degree of freedom).

• Any set of joints can be reduced to these two types.

Links

• $n$ represents the number of moving links plus the base link (fixed link).

• Working with one degree of freedom joints allows connecting them to generalized coordinates.

• A spherical joint has three degrees of freedom and can be represented using three revolute joints with zero link length.

Representing Manipulator Configuration

• The configuration of the manipulator needs to be represented with respect to a fixed frame.

• One approach involves fixing each link individually using multiple vectors (e.g., three vectors at three different points).

  • This requires nine parameters (three parameters per vector in 3D) per link, totaling $9n$ parameters for $n$ links.

Configuration Parameters

• Any set of parameters that fully describes the configuration is a set of configuration parameters.

• A minimal set of independent parameters is desirable.

Generalized Coordinates

• Generalized coordinates are a set of completely independent configuration parameters.

• They are useful for dynamics and directly indicate the number of degrees of freedom of the robot.

Degrees of Freedom

• To determine the number of degrees of freedom, imagine removing the joints, leaving $n$ rigid bodies in space.

• Each rigid body requires six parameters to describe its position and orientation (three for position, three for orientation), totaling $6n$ parameters.

• Placing the joints back introduces constraints. Each one-degree-of-freedom joint introduces five constraints.

  • Revolute joint: three position constraints and two rotation constraints.

  • Prismatic joint: five constraints as well, eliminating two rotational and three position degrees of freedom.

• Total degrees of freedom: $6n - 5n = n$.

• For a manipulator with a fixed base and one-degree-of-freedom joints, the number of degrees of freedom equals the number of joints ($n$).

• For a humanoid robot, if the base (one foot) is fixed, the same logic applies. However, if the base is free to move, there are six additional degrees of freedom (n + 6), though these are not actuated.

End Effector

• The end effector is the last rigid body.

• It has the freedom to position and orient itself.

• A frame attached to the end effector rotates with it, describing its position and orientation.

• The end effector has at most six degrees of freedom, depending on constraints.

End Effector Configuration Parameters

• The configuration of the end effector can be described using configuration parameters or generalized (independent) coordinates (task or operational coordinates).

• With $m$ parameters describing the position and orientation of the end effector with respect to a fixed frame as configuration parameters.

• Examples:

  • Three Euler angles (or fixed angles) for orientation plus X, Y, and Z for position (minimal representation with six parameters).

  • Three vectors to three different points (nine parameters, non-minimal).

  • Euler parameters (four parameters).

  • Direction cosines (nine parameters for orientation) plus position (three parameters), totaling 12 parameters.

Operational Coordinates

• Operational coordinates (or task coordinates) are a set of independent coordinates that describe the position and orientation of the end effector where the task is defined.

• This involves an operational point where the robot is acting.

• If $m_0$ represents the number of independent parameters, it also represents the number of degrees of freedom of the end effector.

• For a robot moving in 3D space with six degrees of freedom, the end effector can be positioned and oriented anywhere (maximum six degrees of freedom).

• For a planar robot, the end effector has three degrees of freedom (X, Y, and one angle).

Redundancy

• If one more joint is added to the robot, the end effector position is fixed, but the robot configuration can vary due to redundancy.

• A robot is redundant if the number of degrees of freedom of the robot $n$ is greater than the number of degrees of freedom of the end effector $\, m_0$.

• Redundancy enables the robot to avoid obstacles. The degree of redundancy is measured by the difference between $\, n$ and $\, m_0$.

Vector Representation

• A point in space $P$ is defined with respect to another point (origin) represented by a vector $\, P$.

• Changing the origin changes the vector.

Coordinate Frames

• Vectors can be expressed with respect to a coordinate frame $(X, Y, Z)$.

• Changing the orientation of the frame changes the components of the vector.

• Transformations are used to relate descriptions in different frames.

• Orthonormal frames are used.

• Transformations must account for rotations and translations of the frame.

Rigid Body Transformation

• A frame $B$ is attached to a rigid body.

• Need to describe frame $B$ with respect to a fixed frame $A$.

• The vector $P_A$ locates the origin of frame $B$ with respect to frame $A$ , described in frame $A$ .

• The orientation of frame $B$ is described by the vectors $X<em>B, Y</em>B$, and $\, Z_B$, also described in frame $A$ .

• The rotation matrix is used to describe these rotations.

Rotation Matrix

• Focusing solely on rotations:

  • The rotation from frame $B$ to frame $A$ is described by a $3 \times 3$ rotation matrix with components $\R<em>{11}, R</em>{12}, R<em>{13}, R</em>{21}, R<em>{22}, R</em>{23}, R<em>{31}, R</em>{32}, R_{33}$.

  • The vector $X_B$ defined in frame $B$ is transformed to its description in frame $A$ using the rotation matrix.

  • Mathematically, this is written as: $X<em>B \text{ defined in frame } B \xrightarrow{\text{Rotation Matrix}} X</em>B \text{ defined in frame } A$

  • Where \XB in frame $B$ is (1, 0, 0), \YB in frame $B$ is (0, 1, 0), and $\Z_B$ in frame $B$ is (0, 0, 1).

  • This implies that the rotation matrix is composed of the components of the axes $X<em>B , Y</em>B , Z_B$ in frame $A$.

  • The rotation matrix from $B$ to $A$ is the component of $X<em>B$ in $A$, the component of $Y</em>B$ in $A$, and the component of $Z_B$ in $A$.

  • The element in the rotation matrix is obtained through the dot product between the vectors of the two frames.

  • $R = \begin{bmatrix} X<em>B \cdot X</em>A &amp; Y<em>B \cdot X</em>A &amp; Z<em>B \cdot X</em>A \ X<em>B \cdot Y</em>A &amp; Y<em>B \cdot Y</em>A &amp; Z<em>B \cdot Y</em>A \ X<em>B \cdot Z</em>A &amp; Y<em>B \cdot Z</em>A &amp; Z<em>B \cdot Z</em>A \end{bmatrix}$

  • The rotation matrix is either the component of $B$ in $A$ or the transpose of the rows of $A$ in $B$.

  • The inverse of the rotation matrix from $B$ to $A$ (i.e., from $A$ to $B$) is simply its transpose: $\R<em>{B \rightarrow A}^{-1} = R</em>{A \rightarrow B} = R_{B \rightarrow A}^T$

Example

• Consider a rotation where $X<em>B$ and $X</em>A$ are aligned, $Y<em>B$ is along the $Z$-axis, and $Z</em>B$ is along the negative $Y$-axis.

• The rotation matrix from $B$ to $A$ is:
  $R = \begin{bmatrix} 1 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; -1 \ 0 &amp; 1 &amp; 0 \end{bmatrix}$

Rigid Body Representation

• A frame $B$ is fully defined with respect to frame $A$ by combining the rotation matrix from $B$ to $A$ with the location of the origin of $B$ with respect to $A$.

Mapping

• A rotation matrix allows describing the same vector (a vector $P$) in two different frames.

• The vector $\P$ is described by its dot product with $A$ to give its component in $A$.

• $P \text{ in } A = R \times P \text{ in } B$

Translation

• Consider frames $A$ and $B$ with the same orientation, but a different origin. The relationship between $\P<em>A$ and $\P</em>B$ for a point P is: $P<em>A = P</em>B + \text{Translation of Origin}$

  • This operation, involving a translation, changes the description from origin $B$ to origin $A$.

General Transformation

• For a point $P$ described with respect to an arbitrary frame $B$ that is rotated with respect to $A$, the transformation is: $P<em>A = R \times P</em>B + \text{Translation of Origin}$

  • The description of $P$ is rotated to frame $A$, and then added to the translation of the origin.

Homogeneous Transformation

• The general transformation can be expressed in a homogeneous form using 4D space: $\begin{bmatrix} P<em>A \ 1 \end{bmatrix} = \begin{bmatrix} R &amp; T \ 0 &amp; 1 \end{bmatrix} \begin{bmatrix} P</em>B \ 1 \end{bmatrix}$

  • This allows representing transformations as matrix multiplications.

Example

• Using the previous rotation matrix and a translation vector of (0, 3, 1):

  • $H = \begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; -1 &amp; 3 \ 0 &amp; 1 &amp; 0 &amp; 1 \ 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}$

  • For point $P$ in frame $B$ described by (0, 1, 1):
    $\begin{bmatrix} P_A \ 1 \end{bmatrix} = \begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; -1 &amp; 3 \ 0 &amp; 1 &amp; 0 &amp; 1 \ 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix} \begin{bmatrix} 0 \ 1 \ 1 \ 1 \end{bmatrix} = \begin{bmatrix} 0 \ 2 \ 2 \ 1 \end{bmatrix}$

  • Dropping the one, the vector is (0, 2, 2).

Operators

• A rotation matrix can be used to rotate a vector.

  • Whereas before it describes a vector in two different frames, it is now performing a rotation in one frame.

• $P<em>2 = R \times P</em>1$

• Rotations about specific axes $(X, Y, Z)$ are of special interest.

• A general rotation rotates a vector $P<em>1$ into a vector $P</em>2$ about an arbitrary vector $\vec{k}$ .

• Example for a rotation of theta degrees about the x-axis:
  $R_x(\theta) = \begin{bmatrix} 1 &amp; 0 &amp; 0 \ 0 &amp; \cos(\theta) &amp; -\sin(\theta) \ 0 &amp; \sin(\theta) &amp; \cos(\theta) \end{bmatrix}$

Translations as Operators

• With mapping, vectors were translated with respect to two different frames. With operators, translation is applied to a point in space.

• Where $\vec{Q}$ is a translation operation:

  $P2 = P1 + \vec{Q}$

• Can be broken down into:
  $\vec{Q} = \begin{bmatrix} Qx \ Qy \ Q_z \ 1 \end{bmatrix}$

• Then we can define the operator as:
  $Q= \begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; Q<em>x \ 0 &amp; 1 &amp; 0 &amp; Q</em>y\ 0 &amp; 0 &amp; 1 &amp; Q_z \ 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}$

General Transformations as Operators Combined

• The general operator that can be imagined is this operator rotating about some vector $\vec{k}$ with some angle theta and translating $\vec{Q}$.

• This can be described as:
  $P2 = \vec{Q} + R \times P1$

  Summarizing:

• The most general form of transformation has 2 components, rotation and translation.

• The transformation can give us Mapping - changing the description; OR

• The transofmration results in changing the vector itself, hence change the point.