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Kinematics: the geometry and description of motion

Kinematics studies how points, bodies and systems move, describing paths, velocity and acceleration without considering forces. It underpins robotics, biomechanics, engineering and astronomy.

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Overview

Kinematics is the branch of mechanics that describes motion without reference to its causes. It focuses on the positions, trajectories and time-dependent changes of points and rigid or flexible bodies. In classical settings it is usually introduced within mechanics as a preliminary to dynamics; historically the word derives from a French adaption of the Greek for movement, a term associated with early work by A. M. Ampère (cinématique). Kinematic analysis treats motion as a geometric and temporal relationship rather than as an explanation that invokes forces.

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Kinematics of Machinery - Figure 21
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Fundamental quantities and representations

Kinematic descriptions use a small set of primary quantities and representations. These include:

  • Position: the location of a point or body in space, often expressed by coordinates relative to a reference frame or geometric object such as a line or plane. See geometric descriptions and the choice of space used to embed motion.
  • Velocity: the rate of change of position with respect to time; a vector that indicates both speed and direction (velocity).
  • Acceleration: the rate of change of velocity; it captures how motion is speeding up, slowing down or changing direction (acceleration).
  • Configuration and constraints: descriptions of how parts of a system are connected, which limit allowable motions and define degrees of freedom.

Mathematical formulation and geometry

Motion can be represented by functions that map time to points in a geometric model. In planar problems rotations are often represented using the unit circle or complex numbers (unit circle, complex plane), while three-dimensional motions make use of vectors, matrices and quaternions. Other algebraic structures express non-Euclidean or relativistic transformations: for example one can model Lorentz transformations that relate spacetime coordinates in special relativity (Lorentz transformations). Mathematicians studying this subject refer to kinematic geometry, in which time (time) is treated as a parameter and motion becomes a parameterized curve or mapping. Rigid transformations — combinations of translations and rotations — are central to describing the movement of interconnected components in machines.

History and development

Early contributors to kinematic thinking include ancient geometers who studied trajectories and mechanisms, while the formal term and systematic study were shaped in the 18th and 19th centuries as mechanics split into kinematics and dynamics. The field evolved as algebra and geometry provided compact ways to describe motion and as engineers and instrument makers posed practical problems that required systematic analysis. Over time kinematics absorbed techniques from algebraic geometry and later from linear algebra and numerical methods to handle complex linkages and multi-body systems.

Applications and examples

Kinematics is widely applied across engineering and science. Examples include:

  • Mechanical linkages and engines: analyzing the allowable motion of connected rods and cams to determine ranges and positions in an engine or mechanism.
  • Robotics: specifying joint angles and trajectories for a robotic arm so the end effector follows a desired path while respecting constraints.
  • Biomechanics: describing human or animal movement by modeling the skeleton and joints (skeleton, human body) and measuring gait, posture and joint kinematics.
  • Astrophysics and celestial mechanics: tracking the positions and velocities of stars, planets and satellites (astrophysics).
  • Design and synthesis: using kinematic synthesis to create mechanisms with prescribed motion ranges and to compute mechanical advantage (mechanical advantage) for specific tasks.

Analysis methods and notable distinctions

Kinematic analysis measures and predicts the motion of systems from given joint inputs or initial conditions; it contrasts with dynamics, which explains motion by invoking forces and masses. Engineers perform forward kinematics (compute positions from joint parameters) and inverse kinematics (determine joint parameters for a desired position) for manipulators and linkages. Kinematics also intersects with algebraic geometry when determining feasible configurations using polynomial constraint equations; the set of allowable positions is often called the configuration space and is explored to avoid collisions and singularities. Practical computation relies on numerical solvers and sensor-based measurement of kinematic quantities.

Further reading and resources

Introductory treatments usually appear in textbooks on mechanics and robotics, and many specialized articles explore planar algebras and advanced representations such as screw theory and dual quaternions. For context and technical development consult resources on mechanical engineering, control and motion planning, or digital libraries that collect historical and modern papers. The terminology and examples above connect to many subfields; for instance, geometric approaches link back to the word roots introduced by Ampère and are discussed in surveys that bridge pure mathematics and applied kinematics (space, rotation, velocity, acceleration, mechanics, history). For engineering practice see materials on design and robotics and for mathematical formulations search treatments that emphasize the unit circle, algebraic methods and the role of complex and relativistic transformations.

Reference systems and coordinate systems

Reference systems form the physical framework in which a movement is described. Coordinate systems are mathematical instruments for their description; however, they are also used outside of physics. In mechanics, the solution of concrete problems always begins with the definition of a reference and coordinate system.

Reference systems

The quantities location, velocity and acceleration depend on the choice of the reference frame.

  • An observer at a platform perceives an arriving train as moving. For a passenger on the train, however, the train is at rest.
  • Observed from Earth, the Sun appears to revolve around the motionless Earth. Viewed from space, the sun is at rest and the earth is moving.

The description of motions is basically possible in all reference systems, but the description differs depending on the reference system. Planetary motion, for example, is much easier to describe with a sun at rest.

A distinction is made between systems at rest, moving and accelerated reference systems, whereby the accelerated ones are a special case of the moving reference systems. The inertial reference frames are of special importance. These are reference frames that are either at rest or moving in a straight line with constant velocity (no rotation and no acceleration), because in inertial frames Newton's first law applies: A force-free body then moves with constant velocity or remains at rest. In accelerated reference systems, on the other hand, apparent forces occur. The earth rotates around its own axis and around the sun; thus, it does not form an inertial frame. For most practical purposes, however, the Earth can be considered to be at rest to a good approximation.

Within the framework of classical mechanics, it is assumed that a location can be assigned to any body at any time. In the framework of quantum mechanics this is no longer possible. There, only residence probabilities can be given. Furthermore, in Classical Mechanics it is assumed that bodies can attain an arbitrarily high velocity and that time passes at the same rate at any location regardless of the motion. Both is not fulfilled in the relativity theory.

Coordinate systems

Coordinate systems are used for the mathematical description of reference systems. Mostly a Cartesiancoordinate system is used, which consists of axes that are perpendicular to each other. It is particularly suitable for describing rectilinear movements. For rotational motions in a plane, polar coordinates are well suited, especially if the origin is the center of the rotational motion. In three-dimensional space, cylindrical coordinates or spherical coordinates are used. If the motion of a vehicle is to be described from the driver's point of view, the accompanying tripod (natural coordinates) is used. The different coordinate systems can be converted with the coordinate transformation. A certain reference system can therefore be described by different coordinate systems.

Polar coordinates
Polar coordinates

Location, velocity, acceleration and jerk

Location, velocity, and acceleration are the three central quantities of kinematics. They are related to each other through time: A change in location over time is velocity, and a change in velocity over time is acceleration. The terms velocity and acceleration refer to a straight direction at any point in time, but this direction can change constantly. For rotational motion, there is instead the angle of rotation, angular velocity, and angular acceleration. All of these quantities are vectors. They not only have a magnitude, but also a direction.

Location

For the location of a point-like body numerous notations are in use: Generally used is {\vec {r}} for the location vector. This points from the coordinate origin to the point in the coordinate system at which the body is located. For Cartesian coordinates, is also xcommon, sometimes xjust stands for the X component of the location vector. If the orbit of the point is known, then the location is also given by the distance traveled salong the orbit. For generalized coordinates, is qcommon. Since the location of a point changes with time , t{\displaystyle {\vec {r}}(t),x(t),s(t)}or also {\vec q}(t)used.

The function that assigns a location to each time is the displacement-time law. In Cartesian coordinates this can be represented by the scalar functions x(t), y(t)and , z(t)which are the components of the location vector:

{\displaystyle t\mapsto {\vec {r}}(t)=\left({\begin{array}{c}x(t)\\y(t)\\z(t)\end{array}}\right)=x(t)\cdot {\vec {e}}_{x}+y(t)\cdot {\vec {e}}_{y}+z(t)\cdot {\vec {e}}_{z}}

where the unit vectors {\displaystyle \{{\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z}\}}represent the basis (vector space) of the Cartesian coordinate system.

Speed

The change in location over time is the velocity v. If the location of a body changes during a time period Δ \Delta tby the distance Δ \Delta s, then during this time period it has the mean velocity

{\displaystyle v={\frac {\Delta s}{\Delta t}}}.

The velocity {\vec v}(t) at any instant in time, the instantaneous velocity, is given by the infinitesimally small change of {\mathrm d}{\vec r}the location vector during the infinitesimally small time period \mathrm {d} t:

{\displaystyle {\vec {v}}(t)=\lim _{\Delta t\to 0}{\frac {{\vec {r}}(t+\Delta t)-{\vec {r}}(t)}{\Delta t}}={\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}={\dot {\vec {r}}}(t)}.

Thus, velocity is the derivative of location with respect to time and is denoted by a point over the location vector. In Cartesian coordinates, the velocity vector has the components v_{x}, v_{y}and , v_{z}zrepresenting respectively the time derivative of the spatial coordinates x, yand

{\displaystyle {\vec {v}}(t)=\left({\begin{array}{c}v_{x}(t)\\v_{y}(t)\\v_{z}(t)\end{array}}\right)=\left({\begin{array}{c}{\frac {\mathrm {d} x}{\mathrm {d} t}}(t)\\{\frac {\mathrm {d} y}{\mathrm {d} t}}(t)\\{\frac {\mathrm {d} z}{\mathrm {d} t}}(t)\end{array}}\right)=\left({\begin{array}{c}{\dot {x}}(t)\\{\dot {y}}(t)\\{\dot {z}}(t)\end{array}}\right)}

The velocity vector {\displaystyle {\vec {v}}=|{\vec {v}}|\cdot {\vec {e}}_{\parallel }} is composed of its magnitude {\displaystyle |{\vec {v}}|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}} and the normalized direction vector {\displaystyle {\vec {e}}_{\parallel }}. This direction vector thereby represents an instantaneous tangent vector to the trajectory {\vec {r}}(t)the particle.

Acceleration

The change in velocity over time is the acceleration a. If the velocity of a body \Delta vchanges \Delta tby the value Δ during a time period Δ , then it has the mean acceleration

{\displaystyle a={\frac {\Delta v}{\Delta t}}}

The acceleration {\vec a}(t)at any instant is given by the infinitesimally small change {\displaystyle \mathrm {d} {\vec {v}}} of the velocity vector during the infinitesimally small period \mathrm {d} t:

{\displaystyle {\vec {a}}(t)=\lim _{\Delta t\to 0}{\frac {{\vec {v}}(t+\Delta t)-{\vec {v}}(t)}{\Delta t}}={\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}={\dot {\vec {v}}}(t)={\ddot {\vec {r}}}(t)}.

Thus, acceleration is the first derivative of velocity with respect to time and is denoted by one point over the velocity vector, and the second derivative of location with respect to time and is denoted by two points over the location vector. In Cartesian coordinates, acceleration is given by its components {\displaystyle a_{x}(t)}, {\displaystyle a_{y}(t)}and {\displaystyle a_{z}(t)}, which are the second time derivatives of the location components x(t), y(t)and : z(t)

{\displaystyle {\vec {a}}(t)=\left({\begin{array}{c}a_{x}(t)\\a_{y}(t)\\a_{z}(t)\end{array}}\right)=\left({\begin{array}{c}{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}(t)\\{\frac {\mathrm {d} ^{2}y}{\mathrm {d} t^{2}}}(t)\\{\frac {\mathrm {d} ^{2}z}{\mathrm {d} t^{2}}}(t)\end{array}}\right)=\left({\begin{array}{c}{\ddot {x}}(t)\\{\ddot {y}}(t)\\{\ddot {z}}(t)\end{array}}\right)}

The acceleration vector {\displaystyle {\vec {a}}={\vec {a}}_{\parallel }+{\vec {a}}_{\perp }}can be separated into two components, which are respectively tangential and normal to the trajectory. The tangential acceleration {\displaystyle {\vec {a}}_{\parallel }}describes the temporal change of the velocity amount |\vec v|and forms a tangent to the trajectory curve:

{\displaystyle {\vec {a}}_{\parallel }={\frac {\mathrm {d} |{\vec {v}}|}{\mathrm {d} t}}\cdot {\vec {e}}_{\parallel }}

The normal acceleration {\displaystyle {\vec {a}}_{\perp }}, on the other hand, describes the change in velocity direction time {\displaystyle {\vec {e}}_{\parallel }}and provides a measure of the curvature of the trajectory:

{\displaystyle {\vec {a}}_{\perp }=|{\vec {v}}|\cdot {\frac {\mathrm {d} {\vec {e}}_{\parallel }}{\mathrm {d} t}}={\frac {|{\vec {v}}|^{2}}{R}}\cdot {\vec {e}}_{\perp }}

where {\displaystyle {\vec {e}}_{\perp }}is a normalized normal vector of the trajectory and Rdenotes the radius of curvature of the trajectory.

Tear

The change in acceleration over time is the jerk j. If the acceleration of a point-like body changes \Delta tby the value Δ a during a time period Δ {\displaystyle \Delta a}, then it has the mean jerk

{\displaystyle j={\frac {\Delta a}{\Delta t}}}

The jerk {\displaystyle {\vec {j}}(t)}at any instant results from the infinitesimally small change {\displaystyle \mathrm {d} {\vec {a}}} in the acceleration vector during the infinitesimally small period \mathrm {d} t:

{\displaystyle {\vec {j}}(t)=\lim _{\Delta t\to 0}{\frac {{\vec {a}}(t+\Delta t)-{\vec {a}}(t)}{\Delta t}}={\frac {\mathrm {d} {\vec {a}}}{\mathrm {d} t}}={\dot {\vec {a}}}(t)={\ddot {\vec {v}}}(t)={\overset {...}{\vec {r}}}(t)}.

Thus, the jerk is the first derivative of acceleration with respect to time and is denoted by two points over the velocity vector, and the third derivative of location with respect to time and is denoted by three points over the location vector. In Cartesian coordinates, the jerk is given by the components {\displaystyle j_{x}(t)}, {\displaystyle j_{y}(t)}and : {\displaystyle j_{z}(t)}

{\displaystyle {\vec {j}}(t)=\left({\begin{array}{c}j_{x}(t)\\j_{y}(t)\\j_{z}(t)\end{array}}\right)=\left({\begin{array}{c}{\dot {a}}_{x}(t)\\{\dot {a}}_{y}(t)\\{\dot {a}}_{z}(t)\end{array}}\right)}

According to this definition, which is mainly used in physics, a uniform circular motion would be a motion with constant jerk. However, in common usage and in engineering applications, this is a jerk-free motion. The acceleration vector is therefore transformed into a body-fixed coordinate system and the derivation is performed in this system. One obtains for the jerk in the body-fixed system:

{\displaystyle {\vec {j}}'(t)={\frac {\mathrm {d} '{\vec {a}}'}{\mathrm {d} t}}},

where {\displaystyle {\vec {a}}'=(T_{K'}^{I})^{-1}\cdot {\vec {a}}}and {\displaystyle T_{K'}^{I}}the transformation matrix from the body fixed system to the inertial system.

In this definition, for example, the lateral pressure, which plays a major role in rail vehicles, is proportional to the change in curvature. For the alignment elements used, this is given analytically as a function of the path and can be converted into the transverse pressure for a specific speed.

Questions and answers

Q: What is kinematics?

A: Kinematics is the branch of classical mechanics which describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without looking at the cause of this motion.

Q: What does kinematic analysis measure?

A: Kinematic analysis measures the kinematic quantities used to describe motion.

Q: What are rigid transformations?

A: Rigid transformations are certain geometric transformations which are used to describe the movement of components in a mechanical system.

Q: How can kinematics be abstracted into mathematical functions?

A: It is possible to represent rotation with elements of the unit circle in the complex plane, and other planar algebras can be used to represent shear mapping in absolute time and space, as well as Lorentz transformations in relativistic space and time.

Q: How can kinematics be applied to engineering?

A: In engineering, kinematic analysis may be used to find the range of movement for a given mechanism, while working in reverse, kinematic synthesis designs a mechanism for a desired range of motion. In addition, it applies algebraic geometry to study mechanical advantage in a mechanical system or mechanism.

Q: Where else is kinematics used besides engineering?

A: Astrophysics uses it to describe celestial body movements and systems; mechanical engineering, robotics and biomechanics use it for joined parts such as an engine or robotic arm; mathematicians have developed a science using time as parameter; and it has been applied to study human skeleton motions.

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