Coordinate systems and transformations are crucial for robots to understand their environment and move effectively. They provide a framework for representing positions, orientations, and movements in space. Different systems, like Cartesian and polar coordinates, suit various situations and geometries.

Transformations allow robots to convert between coordinate systems and perform actions like rotation and translation. Homogeneous coordinates and transformation matrices simplify these operations, enabling robots to plan and execute complex movements accurately. Understanding these concepts is essential for autonomous robot navigation and manipulation.

Types of coordinate systems

• Coordinate systems provide a standardized way to specify the position and orientation of objects in space
• Different types of coordinate systems are used depending on the geometry and symmetry of the problem at hand
• Understanding the properties and transformations between coordinate systems is crucial for autonomous robots to perceive, plan, and act in their environment

Cartesian coordinate system

• Represents points in space using three orthogonal axes: x, y, and z
• Each point is specified by its coordinates $(x, y, z)$, which are the distances from the origin along each axis
• Cartesian coordinates are widely used in computer graphics, CAD, and robotics due to their simplicity and intuitive nature (3D printing, CNC machining)
• However, they may not be the most convenient choice for certain geometries or applications (spherical objects, polar robots)

Polar coordinate system

• Represents points in a plane using a distance from the origin (radius) and an angle from a reference direction (polar angle)
• Each point is specified by its coordinates $(r, \theta)$, where $r$ is the radius and $\theta$ is the polar angle
• Polar coordinates are useful for problems with circular or radial symmetry (radar systems, polar robots)
• They can simplify certain calculations and equations compared to Cartesian coordinates (distance between points, angular relationships)

Cylindrical coordinate system

• Combines elements of Cartesian and polar coordinate systems to represent points in 3D space
• Each point is specified by its coordinates $(r, \theta, z)$, where $r$ and $\theta$ are the polar coordinates in the xy-plane, and $z$ is the height along the z-axis
• Cylindrical coordinates are suitable for problems with axial symmetry (screw threads, cylindrical robots)
• They can simplify calculations involving rotations around the z-axis and translations along it

Spherical coordinate system

• Represents points in 3D space using a distance from the origin (radius), an angle in the xy-plane (azimuth), and an angle from the z-axis (elevation)
• Each point is specified by its coordinates $(r, \theta, \phi)$, where $r$ is the radius, $\theta$ is the azimuth angle, and $\phi$ is the elevation angle
• Spherical coordinates are useful for problems with spherical symmetry (GPS, celestial navigation)
• They can simplify certain calculations and equations related to spherical geometry (distance on a sphere, angular relationships)

Homogeneous coordinates

• Homogeneous coordinates are an extension of Cartesian coordinates that allow the representation of points, vectors, and transformations in a unified manner
• They introduce an additional coordinate, usually denoted as $w$, to the standard Cartesian coordinates $(x, y, z)$
• Homogeneous coordinates enable the use of matrix operations for geometric transformations and projective geometry

Representing points and vectors

• A point in homogeneous coordinates is represented as $(x, y, z, w)$, where $w$ is typically set to 1
• A vector in homogeneous coordinates is represented as $(x, y, z, 0)$, with $w$ set to 0 to indicate the absence of a translation component
• To convert homogeneous coordinates back to Cartesian coordinates, divide $x$, $y$, and $z$ by $w$ (assuming $w \neq 0$)

Advantages of homogeneous coordinates

• Homogeneous coordinates allow the representation of points at infinity (ideal points) by setting $w = 0$
• They enable the use of matrix operations for geometric transformations, such as translation, rotation, scaling, and projection
• Homogeneous coordinates simplify the composition of multiple transformations by representing them as matrix multiplications
• They provide a consistent framework for handling both points and vectors in the same coordinate system

Coordinate transformations

• Coordinate transformations are mathematical operations that map points from one coordinate system to another
• They are essential for expressing the relationship between different frames of reference and for performing geometric manipulations on objects
• Common types of coordinate transformations include translation, rotation, scaling, and shearing

Translation

• Translation is a transformation that moves an object by a specified distance along each coordinate axis
• It is represented by a translation vector $(t_x, t_y, t_z)$, which specifies the displacement along the x, y, and z axes, respectively
• In homogeneous coordinates, translation is performed by adding the translation vector to the coordinates of each point: $(x', y', z', 1) = (x + t_x, y + t_y, z + t_z, 1)$

Rotation

• Rotation is a transformation that rotates an object around a specified axis by a given angle
• It is represented by a rotation matrix, which depends on the axis and angle of rotation
• Common rotation matrices include:
  • Rotation around the x-axis by an angle $\theta$: $R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & \cos\theta & -\sin\theta & 0 \ 0 & \sin\theta & \cos\theta & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$
  • Rotation around the y-axis by an angle $\theta$: $R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta & 0 \ 0 & 1 & 0 & 0 \ -\sin\theta & 0 & \cos\theta & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$
  • Rotation around the z-axis by an angle $\theta$: $R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 & 0 \ \sin\theta & \cos\theta & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$

Scaling

• Scaling is a transformation that changes the size of an object by a specified factor along each coordinate axis
• It is represented by a scaling matrix, which has the scaling factors $(s_x, s_y, s_z)$ along the diagonal: $S(s_x, s_y, s_z) = \begin{bmatrix} s_x & 0 & 0 & 0 \ 0 & s_y & 0 & 0 \ 0 & 0 & s_z & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$
• Scaling can be uniform (same factor along all axes) or non-uniform (different factors along each axis)

Shearing

• Shearing is a transformation that distorts an object by shifting its points along one axis in proportion to their coordinates along another axis
• It is represented by a shearing matrix, which has the shearing factors $(sh_x, sh_y, sh_z)$ in the off-diagonal elements
• Shearing matrices for different axes include:
  • Shearing along the x-axis: $Sh_x(sh_x) = \begin{bmatrix} 1 & sh_x & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$
  • Shearing along the y-axis: $Sh_y(sh_y) = \begin{bmatrix} 1 & 0 & 0 & 0 \ sh_y & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$
  • Shearing along the z-axis: $Sh_z(sh_z) = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & sh_z & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$

Transformation matrices

• Transformation matrices are mathematical objects that represent coordinate transformations in a compact and composable form
• They are square matrices that encode the effect of a transformation on points or vectors in homogeneous coordinates
• Transformation matrices allow the composition of multiple transformations by matrix multiplication, which simplifies the process of applying a sequence of transformations to an object

Matrix representation of transformations

• Translation: A translation by a vector $(t_x, t_y, t_z)$ is represented by the matrix $T(t_x, t_y, t_z) = \begin{bmatrix} 1 & 0 & 0 & t_x \ 0 & 1 & 0 & t_y \ 0 & 0 & 1 & t_z \ 0 & 0 & 0 & 1 \end{bmatrix}$
• Rotation: Rotation matrices for different axes are as described in the previous section
• Scaling: A scaling by factors $(s_x, s_y, s_z)$ is represented by the matrix $S(s_x, s_y, s_z)$ as described in the previous section
• Shearing: Shearing matrices for different axes are as described in the previous section

Composition of transformations

• Transformation matrices can be composed by matrix multiplication to represent a sequence of transformations applied to an object
• The order of multiplication matters, as matrix multiplication is not commutative
• To apply a sequence of transformations $T_1, T_2, \ldots, T_n$ to a point $P$, multiply the transformation matrices in the reverse order: $P' = T_n \cdots T_2 T_1 P$
• Composition allows the creation of complex transformations by combining simpler ones (robot arm movements, computer graphics)

Inverse of transformation matrices

• The inverse of a transformation matrix undoes the effect of the transformation, mapping the transformed points back to their original positions
• The inverse of a transformation matrix is another matrix that, when multiplied with the original matrix, yields the identity matrix
• Inverses are useful for reversing transformations, solving equations, and determining the original coordinates of transformed points
• For rotation and scaling matrices, the inverse is the transpose of the matrix (orthogonal matrices)
• For translation matrices, the inverse is obtained by negating the translation vector
• Shearing matrices have more complex inverses that involve the shearing factors

Euler angles

• Euler angles are a set of three angles that describe the orientation of a rigid body in 3D space
• They represent a sequence of rotations around the coordinate axes, typically in the order of x-y-z (roll, pitch, yaw) or z-y-x (yaw, pitch, roll)
• Euler angles are widely used in robotics, aerospace, and computer graphics to specify the orientation of objects or frames

Roll, pitch, and yaw

• Roll (φ): Rotation around the x-axis, representing the tilting of an object side-to-side
• Pitch (θ): Rotation around the y-axis, representing the tilting of an object forward or backward
• Yaw (ψ): Rotation around the z-axis, representing the turning of an object left or right
• The order of rotations matters, as rotations are not commutative
• Different conventions exist for the order and names of the angles (x-y-z, z-y-x, intrinsic, extrinsic)

Gimbal lock problem

• Gimbal lock is a singularity that occurs when two of the three rotation axes align, resulting in a loss of one degree of freedom
• It happens when the pitch angle (θ) approaches ±90°, causing the roll and yaw axes to become parallel
• In this situation, changes in roll and yaw produce the same effect, making it impossible to distinguish between them
• Gimbal lock can cause problems in applications that rely on smooth and unambiguous rotations (robotics, 3D graphics)
• Quaternions or other rotation representations can be used to avoid gimbal lock

Quaternions

• Quaternions are a four-dimensional extension of complex numbers that can be used to represent rotations in 3D space
• They consist of a scalar part (w) and a vector part (x, y, z), written as $q = w + xi + yj + zk$, where $i$, $j$, and $k$ are imaginary units
• Quaternions provide a compact and numerically stable way to represent and compose rotations without suffering from gimbal lock

Representation of rotations

• A rotation by an angle $\theta$ around a unit vector $\vec{u} = (u_x, u_y, u_z)$ can be represented by a quaternion $q = (\cos\frac{\theta}{2}, \vec{u}\sin\frac{\theta}{2})$
• The scalar part $w = \cos\frac{\theta}{2}$ represents the amount of rotation, while the vector part $(x, y, z) = \vec{u}\sin\frac{\theta}{2}$ represents the axis of rotation
• Quaternions can be normalized to have a magnitude of 1, which ensures that they represent valid rotations
• Composition of rotations is achieved by quaternion multiplication, which is non-commutative

Advantages over Euler angles

• Quaternions avoid gimbal lock, as they do not suffer from singularities like Euler angles
• They provide a smooth and continuous representation of rotations, without discontinuities or ambiguities
• Quaternions are more numerically stable and efficient for interpolation and integration of rotations (spherical linear interpolation, quaternion averaging)
• They can be easily converted to and from rotation matrices and other rotation representations
• Quaternions are widely used in computer graphics, virtual reality, and robotics for orientation representation and control

Forward vs inverse kinematics

• Kinematics is the study of the motion of objects without considering the forces that cause the motion
• In robotics, kinematics deals with the relationship between the joint angles and the position and orientation of the end-effector (tool or device at the end of a robot arm)
• Forward and inverse kinematics are two fundamental problems in robotics that involve the mapping between joint angles and end-effector pose

Forward kinematics

• Forward kinematics (FK) is the process of determining the position and orientation of the end-effector given the joint angles of the robot arm
• It involves applying a sequence of coordinate transformations from the base frame to the end-effector frame, using the known joint angles and link lengths
• FK is a straightforward problem with a unique solution, as the end-effector pose is uniquely determined by the joint angles
• It is used for visualization, collision detection, and control of robot arms

Inverse kinematics

• Inverse kinematics (IK) is the process of determining the joint angles required to achieve a desired position and orientation of the end-effector
• It involves solving a set of nonlinear equations that relate the end-effector pose to the joint angles, which can be challenging and computationally expensive
• IK may have multiple solutions (redundant robots) or no solutions (unreachable poses), depending on the robot's structure and constraints
• It is used for motion planning, trajectory generation, and high-level control of robot arms

Applications in robotics

• Forward and inverse kinematics are essential for the control and simulation of robot arms in various applications:
  • Industrial robotics: Manipulating objects, welding, painting, assembly
  • Medical robotics: Surgical assistance, rehabilitation, prosthetics
  • Service robotics: Household tasks, personal assistance, entertainment
  • Space robotics: Spacecraft maintenance, asteroid mining, planetary exploration
• Efficient and accurate solutions to FK and IK problems are crucial for the performance and safety of robotic systems

Coordinate frames in robotics

• Coordinate frames are local reference systems attached to different parts of a robot or its environment
• They provide a consistent way to describe the position and orientation of objects and to perform coordinate transformations between different frames
• Common coordinate frames in robotics include the base frame, end-effector frame, and intermediate frames

Base frame

• The base frame (or world frame) is a fixed reference frame attached to the base of the robot or its environment
• It serves as the global coordinate system in which the robot's position and orientation are defined
• The base frame is usually chosen to coincide with a convenient and stable reference point (robot's base, workbench, room corner)
• All other frames are described relative to the base frame using coordinate transformations

End-effector frame

• The end-effector frame (or tool frame) is a local reference frame attached to the end-effector of the robot arm
• It describes the position and orientation of the tool or device mounted on the robot's end-effector
• The end-effector frame is important for specifying the desired pose of the tool and for planning and executing manipulation tasks
• The relationship between the end-effector