(Reference) Frame

As defined in ISO 19111, a reference frame is a parameter or set of parameters that realize the position of the origin, the scale, and the orientation of a coordinate system.

Another, more practical definition from Wikipedia is: "In physics, a (reference) frame consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame."

A reference frame can be visualized as a subspace with its own coordinate system (represented with an arrow for each axis and a 2D grid to facilitate measurements) and a point of origin (represented as a sphere at the point where the arrows intersect). Generally, to simplify calculations, the associated coordinate systems are Cartesian in nature, but in the case of non-euclidean spaces, they require more advanced definitions (like the geographical coordinate systems used in global positioning systems).

Pose

"A pose determines the position and orientation of an object (or group of objects) relative to a coordinate system.”

In robotics and human-computer interaction, this concept is often referred to as Six Degrees of Freedom (6DoF) and it is used to specify the position and orientation of an object with great precision (generally, by combining a Local Positioning System for the translation and an Aircraft Principal Axes Orientation System for the rotation). In this way, it is possible to define complex kinematic systems that allow the creation of advanced robotic arms and exoskeletons.

From the point of view of computer graphics, however, a pose can be seen as a simplified version of a "transform class" (without the scale-related fields/properties). This direct correspondence implies that a pose is associated to a single node of an scene graph (i.e., the "hand" has a pose, the "arm" has another pose, etc.), but, when 3D artists and engineers talk about "poses", they are generally referring to the combined pose data of all the components of an entity (i.e., the pose of a dummy, as seen in the illustration). In fact, most 3D editing software solutions include a separate "pose mode" so that artists can independently work on the geometry of the different 3D models and on their animation.

Pose Chain

“A pose chain is a sequence of poses that is used to determine the position and orientation of objects across multiple frames.”

When dealing with multiple reference frames (specially, when they are of different types), it is necessary to perform several complex transformation operations to convert between the different Coordinate Reference Systems. To facilitate this time-consuming process and to ensure that the operations are applied in the right order, the different poses are often organized into one-dimensional sequences of poses called "chains".

In many applications, where a large number of entities have similar pose chains, it is common to reorganize this structure into a graph, so that it is possible to more effectively apply the transformation operations (by storing the result of the repeated operations). In computer graphics, this concept is the basis of the "scene graph", in which the transform class instances of the different entities are connected together using a hierarchical graph through (single) parent-children relationships. In this way, the pose of each entity of the graph is calculated in a depth-first manner that minimizes the number of required calculations.

OGC GeoPose

"A geographically-anchored pose is a pose defined in relation to a real-world object (by default, Earth), typically, via a geographical Reference frame."

When a "GeoPose" is provided, it will be understood to mean a geographically-anchored pose that complies with the OGC GeoPose specification. The goal of the standardization of OGC GeoPose is to establish a single mechanism to encode the position and orientation of any real/virtual objects within a geographical (ellipsoidal) frame, greatly simplifying the determination of the pose of entities relative to an astronomical object and, consequently, relative to each other.

In practical terms, a GeoPose facilitates the application of one or more transformation operations to properly position and orientate modes in a scene graph (generally, the root one) on a realistic 3D map and/or in an Augmented/Mixed Reality environment.

Location

"Not to be confused with the term position, a location is a distinctive region of a space."

While a position merely specifies a single point within a space (relative to a frame), a location defines a limited region within a space (often with its very own frame). A location generally receives a unique name within a context (or within another location) to easily identify it from others and, if necessary, it can also have a additional semantic information to further differentiate it (e.g., "Washington D.C.", "Washington State" and "Washington City, Utah"). There are as many types of locations as there are types of spaces (e.g., "the bottom of screen", "behind the car", "Earth’s orbit", etc.), but it is specially relevant in the context of geography, where the geolocation ("In which street/road/town I am?") is often much more important than the actual geoposition ("what are my GPS coordinates?").

The boundaries of locations are often defined using either dimensional properties (i.e., width, height and depth) or specific shapes (most notably, 2D projections in a geographical space called geofences). However, when there are a large number of locations or these are constantly changing, the boundaries are defined by proximity to the closest point in the topological skeleton or by the minimum number of logical connections.

Local Systems

"A Local Positioning System defines the location of objects relative to a frame based on Euclidean spaces."

These frames are often defined with an arbitrary point of origin depending on the nature of the subspace to represent. When working with physical spaces or virtual 3D scenes, the point of origin is generally situated in the center of the space (to avoid the limitations of single-precision floating-point numbers). However, when operating within 2D spaces with clear boundaries, like screens or documents, the point of origin is normally placed in the top-left corner (due to the graphic systems based on the left-to-right, top-to-bottom writing direction used in most western countries). Needless to say, this makes the transformations between the different frames (i.e., determining what 3D object has the user selected on a 2D screen) a very cumbersome process.

In recent years, advancements in Indoor Positioning Systems technologies have enabled the precise tracking of real-world elements in relatively large local subspaces. Yet, when a use case has more than a single location or the curvature of the Earth becomes an important factor to take into account, using a Global Location System is generally a better option.

Global Systems

"A Global Positioning System defines the location of objects relative to a frame based on spherical spaces, generally associated with the planet Earth."

As its name implies, a Global Positioning System is defined by a spherical object that establishes the properties of a frame itself (even if the astronomical body is not a perfect sphere). The resulting subspace is delimited in the two horizontal dimensions of the surface of a sphere (longitude should only have values between -180 and 180 degrees, whereas the latitude values should be within a -90 to 90 degrees) and partially delimited in the vertical dimension (altitude values should never be lower than the negative value of the radius of the sphere). As for the point of origin of this subspace, for practical reasons, it is generally placed on an arbitrary point alongside the equator of the sphere (thus, also defining an offset value for the altitude values).

Traditionally, determining the position of an object on the surface of the Earth with any degree of accuracy required cumbersome tools and manual calculations. Fortunately, satellite-based radio-navigation systems like GPS, Galileo or NavIC have greatly simplified the process and -as long as the devices can establish visual contact with three satellites- it is possible to track the position of all kinds of objects on Earth.

Universal Systems

"A Universal Positioning System defines the location of objects relative to an inertial frame based on Minkowski spaces."

These systems aim to provide a more precise and truly unique positioning for objects relative to an inertial point in the physical Universe (generally, the barycenter of the Solar System). However, since the Minkowski spaces are defined by the rules of Special Relativity, the very subspaces are "curved" by Lorentz Transformations and any traversal is limited by the Speed of Light.

Needless to say, the extremely complex calculations that these positioning systems require relegate them to applications related with interplanetary travel and communications. For example, the SPICE framework created by the NASA’s Planetary Science Division operates with both the J2000 and ICRF Inertial Frames to calculate transfer orbits between bodies of the Solar System.

Euler Angles

"The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system." (from Wikipedia)

The angles are generally specified in degrees (in a similar way to longitude/latitude values in Global Positioning Systems), although the internal operations on computers are almost always performed in radians, one axis after another. Usually, the application order of rotation transformations is the inverse of the specification (the Z angle is applied first, then the Y on, and finally, the X one), but it might vary depending on the use case or the hand rule employed.

It is important to note that, since the rotation operations are applied globally and linked together by the application order, if two axes are driven into a parallel configuration, it can generate a Gimbal Lock, resulting in the loss of one degree of freedom. This is usually not a problem when defining simple, discontinuous orientations, but when the use case involves overlapping or interpolating between multiple Euler angles, the result can be negatively affected.

Look-At Systems

"A Look-At system enables the definition of orientation relative to another object or position vector."

Similar to Quaternions, Look-At systems use a vector (obtained from the difference of the target position minus the current position) to determine the orientation of an object. After normalizing the vector, it is possible to use its components to create a quaternion and apply the same mathematical operations. However, to properly determine the rotation angle around the axial vector (the real component of a quaternion), it is necessary to also provide an "up" vector to serve as a pivot.

Aircraft Principal Axes

"Aircraft Principal Axes define the relative (local) rotation of an object, using a plane flying in the +X axis as a reference."

This system defines the orientation of an object as a combination of rotations along the three axes of an aircraft: yaw (or normal) axis, pitch (or transverse) axis and roll (or longitudinal) axis. These axes have a direct correspondence with the X, Y and Z axis of the (right-handed) Euler system and are defined with the same units. However, since these axes are applied locally (i.e., rotating the coordinate system of the object at each step), the order of application becomes irrelevant.

The Aircraft Principal Axes system has the advantage of being very intuitive (once the users fully understand what axis is related to the terms yaw, pitch and roll) and of not being susceptible to Gimbal Lock issues. On the other hand, this orientation system often requires additional computational power to generate the intermediate rotation matrices (one for each axis with a value different from 0) and, while the image of an airplane can be projected onto other vehicles, it might be difficult to do so with other static elements (e.g., it doesn’t make sense to define the "roll angle" of buildings or any other static structures).

Quaternions

"In computer graphics, a Quaternion is 4D complex number that is used to specify orientations in a 3D space."

In essence, a Quaternion is a mathematical construct with 3 imaginary components (i, j and k) and one real component. When it is used to describe 3D orientations, each imaginary component describes the dimension of a unitary vector in a particular axis of the space (i, j and k are assigned to the X, Y and Z axis, respectively) whereas the real component expresses half of the rotation around that unitary vector, expressed in radians. However, to ensure that the resulting axial vector is unitary (i.e., its length equals one), a quaternion has to follow the fundamental formula i^2 = j^2 = k^2 = i⋅j⋅k = -1.

Due to their mathematical construction and the possibility to interpolate them both linearly and spherically, most 3D engines use Quaternions internally to orientate and animate compound objects. However, the complex nature and the interdependence between imaginary components, make Quaternions very difficult for most human beings to understand and use, so they are rarely employed in user interfaces or human-readable documents.

Animation

"Animation is a method to create the illusion of movement through a collection of still images."

In computer graphics, animation is generally simulated by modifying the properties of an object (including its position, rotation, scale, material colors, etc.) over a period of time (or timeline). Since complex deformations require a large number of physics calculations, most animation systems either operate with properties independently (i.e, separating position, orientation and scale) or define skeleton-based systems to alter the geometry of the object in a controllable way. In recent years, however, many 3D engines incorporate procedural subsystems to simulate advanced animations (such as explosions, wind systems, collision deformations, etc) in real-time, with relatively high degree of precision.

Internally, each change in properties is stored into an "animation key," and when there is more than one key over time, an "animation curve" is generated to specify how to interpolate between them. A collection of curves is called an "animation clip" and can be associated to a given action or event.

Motion

"A motion is a type of animation that only defines the movement (position+orientation keys) of objects."

In many use cases, it is possible to simplify complex animations by segmenting an object (generally, using a skeleton structure) and reducing the types of keys to just position and orientation (creating animation curves known as Motion Paths). This idea of only taking into consideration the movement of different parts of an object is the basis of motion capture techniques; a series of technologies that can even record subtle facial expressions of human beings.

In addition, motions can also be defined as transitions between different poses of an object. In this way, instead of having to define a single sequence of animation keys, it is possible to "jump" between different poses by merely interpolating between them whenever necessary (something especially useful when dealing with real-time camera -motion- tracking systems).

It is important to note that a motion between two poses also generates a direction vector and an attitude (orientation relative to that direction vector).

Trajectory

"A trajectory is a precalculated motion path that attempts to describe the future movement of an object."

While motion paths describe the actual movements over time, in many use cases it is imperative to also be able to predict the future movement of an object (e.g., to avoid collisions or reach a location/orbit with a very specific speed and orientation). These special motion paths are called trajectories and are usually calculated by simply taking into consideration the inertia of an object.

This is especially important for objects placed in stable trajectories relative to others objects (usually, astronomical bodies), because if those trajectories are entirely predictable, it is possible to create an Ephemeris system to rapidly calculate their position at any point in time. Additionally, if a predictable trajectory is regularly repeated over time, it can also be called a stable orbit.

Gesture

"A gesture is a pose or motion used to convey a non-vocal message."

Gestures are a form of communication that involve the positioning/movement of a part of the body in a predetermined way to express a particular concept. And although this type of communication is used by many living beings, humans have used them to develop complete sign languages or to serve as a method of identification. Some basic gestures even allow people to overcome linguistic barriers and create a culture around specific gestures.

Nevertheless, since messages conveyed by gestures require both the issuer to be able to make the gestures and the receiver to be able to interpret them correctly, both the communication channel (the physical space) and the meaning of each gesture has to be previously defined to avoid misunderstandings.

It is important to clarify that gestures can also be conveyed by a visual representations (e.g., the Unicode symbol U+1F44C "👌") or with robotic systems that emulate the body structure and movements of the living beings (which is one of the basis of the animal-robot interaction field).