Kinematic

In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.

Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems. These systems may be rectangular like cartesian, Curvilinear coordinates like polar coordinates or other systems. The object trajectories may be specified with respect to other objects which may themselve be in motion relative to a standard reference. Rotating systems may also be used.

Numerous practical problems in kinematics involve constraints, such as mechanical linkages, ropes, or rolling disks.

Overview

Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.
Kinematics differs from ''dynamics'' (also known as ''kinetics'') which studies the effect of forces on bodies.

Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined.

Another way to describe kinematics is as the specification of the possible states of a physical system. Dynamics then describes the evolution of a system through such states. Robert Spekkens argues that this division cannot be empirically tests and thus has no physical basis.

Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering, robotics, and biomechanics, kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the human skeleton.

Geometric transformations, including called rigid transformations, are used to describe the movement of components in a mechanical system, simplifying the derivation of the equations of motion. They are also central to dynamic analysis.

Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion.J. M. McCarthy and G. S. Soh, 2010
''Geometric Design of Linkages,''
Springer, New York. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism.

Relativistic kinematics applies the special theory of relativity to the geometry of object motion. It encompasses time dilation, length contraction and the Lorentz transformation. The kinematics of relativity operates in a spacetime geometry where spatial points are augmented with a time coordinate to form 4-vectors.

Werner Heisenberg reinterpreted classical kinetics for quantum systems in his 1925 paper "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships". Dirac noted the similarity in structure between Heisenberg's formulations and classical Poisson brackets. In a follow up paper in 1927 Heisenberg showed that classical kinematic notions like velocity and energy are valid in quantum mechanics, but pairs of conjugate kinematic and dynamic quantities cannot be simultaneously measure, a result he called indeterminacy, but which became known as the uncertainty principle.

Etymology

The term kinematic is the English version of A.M. Ampère's ''cinématique'', which he constructed from the Greek ''kinema'' ("movement, motion"), itself derived from ''kinein'' ("to move").

Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek ''grapho'' ("to write").

Kinematics of a particle trajectory in a non-rotating frame of reference

Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the ''x''-axis and north is in the direction of the ''y''-axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If the tower is 50 m high, and this height is measured along the ''z''-axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m).

In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to a reference frame.

The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector $$ can be expressed as
$\mathbf r = (x,y,z) = x\hat\mathbf x + y\hat\mathbf y + z\hat\mathbf z,$
where $x$, $y$, and $z$ are the Cartesian coordinates and $\hat\mathbf x$, $\hat\mathbf y$ and $\hat\mathbf z$ are the unit vectors along the $x$, $y$, and $z$ coordinate axes, respectively. The magnitude of the position vector $\left, \mathbf r\$ gives the distance between the point $\mathbf r$ and the origin.
$, \mathbf, = \sqrt.$
The direction cosines of the position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector.

The ''trajectory'' of a particle is a vector function of time, $\mathbf(t)$, which defines the curve traced by the moving particle, given by
$\mathbf r(t) = x(t)\hat\mathbf x + y(t) \hat\mathbf y +z(t) \hat\mathbf z,$
where $x(t)$, $y(t)$, and $z(t)$ describe each coordinate of the particle's position as a function of time.

Velocity and speed

The velocity of a particle is a vector quantity that describes the ''direction'' as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle (displacement) by the time interval. This ratio is called the average velocity over that time interval and is defined as$\mathbf\bar v = \frac = \frac\hat\mathbf x + \frac\hat\mathbf y + \frac\hat\mathbf z =\bar v_x\hat\mathbf x + \bar v_y\hat\mathbf y + \bar v_z \hat\mathbf z \,$where $\Delta \mathbf$ is the displacement vector during the time interval $\Delta t$. In the limit that the time interval $\Delta t$ approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector,
$\mathbf v = \lim_\frac = \frac = v_x\hat\mathbf x + v_y\hat\mathbf y + v_z \hat\mathbf z .$
Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.

The speed of an object is the magnitude of its velocity. It is a scalar quantity:
$v=, \mathbf, = \frac ,$
where $s$ is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, $\frac$ is non-negative, which implies that speed is also non-negative.

Acceleration

The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio.
$\mathbf\bar a = \frac = \frac\hat\mathbf x + \frac\hat\mathbf y + \frac\hat\mathbf z =\bar a_x\hat\mathbf x + \bar a_y\hat\mathbf y + \bar a_z \hat\mathbf z \,$
where Δv is the average velocity and Δ''t'' is the time interval.

The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative,
$\mathbf a = \lim_\frac =\frac = a_x\hat\mathbf x + a_y\hat\mathbf y + a_z \hat\mathbf z .$

Alternatively,
$\mathbf a = \lim_\frac = \frac = a_x\hat\mathbf x + a_y\hat\mathbf y + a_z \hat\mathbf z .$

Thus, acceleration is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.

The magnitude of the acceleration of an object is the magnitude , a, of its acceleration vector. It is a scalar quantity:
$, \mathbf, = , \dot , = \frac.$

Relative position vector

A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points.
The position of one point ''A'' relative to another point ''B'' is simply the difference between their positions

:$\mathbf_ = \mathbf_ - \mathbf_$

which is the difference between the components of their position vectors.

If point ''A'' has position components $\mathbf_ = \left( x_, y_, z_ \right)$

and point ''B'' has position components $\mathbf_ = \left( x_, y_, z_ \right)$

then the position of point ''A'' relative to point ''B'' is the difference between their components: $\mathbf_ = \mathbf_ - \mathbf_ = \left( x_ - x_, y_ - y_, z_ - z_ \right)$

Relative velocity

The velocity of one point relative to another is simply the difference between their velocities
$\mathbf_ = \mathbf_ - \mathbf_$
which is the difference between the components of their velocities.

If point ''A'' has velocity components $\mathbf_ = \left( v_, v_, v_ \right)$ and point ''B'' has velocity components $\mathbf_ = \left( v_, v_, v_ \right)$ then the velocity of point ''A'' relative to point ''B'' is the difference between their components:
$\mathbf_ = \mathbf_ - \mathbf_ = \left( v_ - v_, v_ - v_, v_ - v_ \right)$

Alternatively, this same result could be obtained by computing the time derivative of the relative position vector r_B/A.

Relative acceleration

The acceleration of one point ''C'' relative to another point ''B'' is simply the difference between their accelerations.
$\mathbf_ = \mathbf_ - \mathbf_$
which is the difference between the components of their accelerations.

If point ''C'' has acceleration components $\mathbf_ = \left( a_, a_, a_ \right)$
and point ''B'' has acceleration components $\mathbf_ = \left( a_, a_, a_ \right)$
then the acceleration of point ''C'' relative to point ''B'' is the difference between their components: $\mathbf_ = \mathbf_ - \mathbf_ = \left( a_ - a_ , a_ - a_ , a_ - a_ \right)$

Assuming that the initial conditions of the position, $\mathbf_0$, and velocity $\mathbf_0$ at time $t = 0$ are known, the first integration yields the velocity of the particle as a function of time.
$\mathbf(t) = \mathbf_0 + \int_0^t \mathbf(\tau) \, \text\tau$

Additional relations between displacement, velocity, acceleration, and time can be derived. If the acceleration is constant,
$\mathbf = \frac = \frac$ can be substituted into the above equation to give:
$\mathbf(t) = \mathbf_0 + \left(\frac\right) t .$

A relationship between velocity, position and acceleration without explicit time dependence can be obtained by solving the average acceleration for time and substituting and simplifying

$t = \frac$

$\left(\mathbf - \mathbf_0\right) \cdot \mathbf = \left( \mathbf - \mathbf_0 \right) \cdot \frac \ ,$
where $\cdot$ denotes the dot product, which is appropriate as the products are scalars rather than vectors.
$2 \left(\mathbf - \mathbf_0\right) \cdot \mathbf = , \mathbf, ^2 - , \mathbf_0, ^2.$

The dot product can be replaced by the cosine of the angle between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case:
$2 \left, \mathbf - \mathbf_0\ \left, \mathbf\ \cos \alpha = , \mathbf, ^2 - , \mathbf_0, ^2.$

In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors () is 0, so $\cos 0 = 1$, and
$, \mathbf, ^2= , \mathbf_0, ^2 + 2 \left, \mathbf\ \left, \mathbf-\mathbf_0\.$
This can be simplified using the notation for the magnitudes of the vectors $, \mathbf, =a, , \mathbf, =v, , \mathbf-\mathbf_0, = \Delta r$ where $\Delta r$ can be any curvaceous path taken as the constant tangential acceleration is applied along that path, so
$v^2= v_0^2 + 2a \Delta r.$
This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that $\Delta r = \int v \, \textt$ or $\Delta r$ is the area under a velocity–time graph. We can take $\Delta r$ by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the $A \cdot B$ where $A$ is the width and $B$ is the height. In this case $A = t$ and $B = v_0$ (the $A$ here is different from the acceleration $a$). This means that the bottom area is $tv_0$. Now let's find the top area (a triangle). The area of a triangle is $\frac BH$ where $B$ is the base and $H$ is the height. In this case, $B = t$ and $H = at$ or $A = \frac BH = \frac att = \frac at^2 = \frac$. Adding $v_0 t$ and $\frac$ results in the equation $\Delta r$ results in the equation $\Delta r = v_0 t + \frac$. This equation is applicable when the final velocity is unknown.

Particle trajectories in cylindrical-polar coordinates

It is often convenient to formulate the trajectory of a particle r(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t'')) using polar coordinates in the ''X''–''Y'' plane. In this case, its velocity and acceleration take a convenient form.

Recall that the trajectory of a particle ''P'' is defined by its coordinate vector r measured in a fixed reference frame ''F''. As the particle moves, its coordinate vector r(''t'') traces its trajectory, which is a curve in space, given by:
$\mathbf r(t) = x(t)\hat\mathbf x + y(t) \hat\mathbf y +z(t) \hat\mathbf z,$
where x̂, ŷ, and ẑ are the unit vectors along the ''x'', ''y'' and ''z'' axes of the reference frame ''F'', respectively.

Consider a particle ''P'' that moves only on the surface of a circular cylinder ''r''(''t'') = constant, it is possible to align the ''z'' axis of the fixed frame ''F'' with the axis of the cylinder. Then, the angle ''θ'' around this axis in the ''x''–''y'' plane can be used to define the trajectory as,
$\mathbf(t) = r\cos(\theta(t))\hat\mathbf x + r\sin(\theta(t))\hat\mathbf y + z(t)\hat\mathbf z,$
where the constant distance from the center is denoted as ''r'', and ''θ''(''t'') is a function of time.

The cylindrical coordinates for r(''t'') can be simplified by introducing the radial and tangential unit vectors,
$\hat\mathbf r = \cos(\theta(t))\hat\mathbf x + \sin(\theta(t))\hat\mathbf y, \quad \hat\mathbf\theta = -\sin(\theta(t))\hat\mathbf x + \cos(\theta(t))\hat\mathbf y .$
and their time derivatives from elementary calculus:
$\frac = \omega\hat\mathbf\theta .$
$\frac = \frac = \alpha\hat\mathbf\theta - \omega^2\hat\mathbf r .$

$\frac = -\omega\hat\mathbf r .$
$\frac = \frac = -\alpha\hat\mathbf r - \omega^2\hat\mathbf\theta.$

Using this notation, r(''t'') takes the form,
$\mathbf(t) = r\hat\mathbf r + z(t)\hat\mathbf z .$
In general, the trajectory r(''t'') is not constrained to lie on a circular cylinder, so the radius ''R'' varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes:
$\mathbf(t) = r(t)\hat\mathbf r + z(t)\hat\mathbf z .$
Where ''r'', ''θ'', and ''z'' might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector v_''P'' is the time derivative of the trajectory r(''t''), which yields:
$\mathbf v_P = \frac \left(r\hat\mathbf r + z \hat\mathbf z \right) = v\hat\mathbf r + r\mathbf\omega\hat\mathbf\theta + v_z\hat\mathbf z = v(\hat\mathbf r + \hat\mathbf\theta) + v_z\hat\mathbf z .$

Similarly, the acceleration a_''P'', which is the time derivative of the velocity v_''P'', is given by:
$\mathbf_P = \frac \left(v\hat\mathbf r + v\hat\mathbf\theta + v_z\hat\mathbf z\right) =(a - v\omega)\hat\mathbf r + (a + v\omega)\hat\mathbf\theta + a_z\hat\mathbf z .$

The term $-v\omega\hat\mathbf r$ acts toward the center of curvature of the path at that point on the path, is commonly called the centripetal acceleration. The term $v\omega\hat\mathbf\theta$ is called the Coriolis acceleration.

Constant radius

If the trajectory of the particle is constrained to lie on a cylinder, then the radius ''r'' is constant and the velocity and acceleration vectors simplify. The velocity of v_P is the time derivative of the trajectory r(''t''),
$\mathbf v_P = \frac \left(r\hat\mathbf r + z \hat\mathbf z \right) = r\omega\hat\mathbf\theta + v_z\hat\mathbf z = v\hat\mathbf\theta + v_z\hat\mathbf z .$

Planar circular trajectories

A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the ''z'' axis:
$\mathbf r(t) = r\hat\mathbf r + z \hat\mathbf z,$
where ''r'' and ''z''₀ are constants. In this case, the velocity v_''P'' is given by:
$\mathbf_P = \frac \left(r\hat\mathbf r + z \hat\mathbf z\right) = r\omega\hat\mathbf\theta = v\hat\mathbf\theta,$
where $\omega$ is the angular velocity of the unit vector around the ''z'' axis of the cylinder.

The acceleration a_''P'' of the particle ''P'' is now given by:
$\mathbf_P = \frac = a\hat\mathbf\theta - v\theta\hat\mathbf r.$

The components
$a_r = - v\theta, \quad a_ = a,$
are called, respectively, the ''radial'' and ''tangential components'' of acceleration.

The notation for angular velocity and angular acceleration is often defined as
$\omega = \dot, \quad \alpha = \ddot,$
so the radial and tangential acceleration components for circular trajectories are also written as
$a_r = - r\omega^2, \quad a_ = r\alpha.$

Point trajectories in a body moving in the plane

The movement of components of a mechanical system are analyzed by attaching a reference frame to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used to define this relative movement. This reduces the description of the motion of the various parts of a complicated mechanical system to a problem of describing the geometry of each part and geometric association of each part relative to other parts.

Geometry is the study of the properties of figures that remain the same while the space is transformed in various ways—more technically, it is the study of invariants under a set of transformations. These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry.

The coordinates of points in a plane are two-dimensional vectors in R² (two dimensional space). Rigid transformations are those that preserve the distance between any two points. The set of rigid transformations in an ''n''-dimensional space is called the special Euclidean group on R^''n'', and denoted SE(''n'').

Displacements and motion

The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say ''M'', on one that moves relative to a fixed frame, ''F,'' on the other. The rigid transformation, or displacement, of ''M'' relative to ''F'' defines the relative position of the two components. A displacement consists of the combination of a rotation and a translation.

The set of all displacements of ''M'' relative to ''F'' is called the configuration space of ''M.'' A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the motion of ''M'' relative to ''F.'' The motion of a body consists of a continuous set of rotations and translations.

Matrix representation

The combination of a rotation and translation in the plane R² can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2 rotation matrix ''A''(''φ'') and the 2×1 translation vector d = (''d_x'', ''d_y''), as:
(\phi, \mathbf)= \begin A(\phi) & \mathbf \\ \mathbf 0 & 1\end
= \begin \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end.
These homogeneous transforms perform rigid transformations on the points in the plane ''z'' = 1, that is, on points with coordinates r = (''x'', ''y'', 1).

In particular, let r define the coordinates of points in a reference frame ''M'' coincident with a fixed frame ''F''. Then, when the origin of ''M'' is displaced by the translation vector d relative to the origin of ''F'' and rotated by the angle φ relative to the x-axis of ''F'', the new coordinates in ''F'' of points in ''M'' are given by:
\mathbf = (\phi, \mathbf)mathbf
= \begin \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end \beginx\\y\\1\end.

Homogeneous transforms represent affine transformations. This formulation is necessary because a translation is not a linear transformation of R². However, using projective geometry, so that R² is considered a subset of R³, translations become affine linear transformations.

Pure translation

If a rigid body moves so that its reference frame ''M'' does not rotate (''θ'' = 0) relative to the fixed frame ''F'', the motion is called pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory d(''t'') of the origin of ''M,'' that is:
\mathbf(t)= (0,\mathbf(t))\mathbf = \mathbf(t) + \mathbf.

Thus, for bodies in pure translation, the velocity and acceleration of every point ''P'' in the body are given by:
$\mathbf_P=\dot(t) = \dot(t)=\mathbf_O, \quad \mathbf_P=\ddot(t) = \ddot(t) = \mathbf_O,$
where the dot denotes the derivative with respect to time and v_''O'' and a_''O'' are the velocity and acceleration, respectively, of the origin of the moving frame ''M''. Recall the coordinate vector p in ''M'' is constant, so its derivative is zero.

Rotation of a body around a fixed axis

Objects like a playground merry-go-round, ventilation fans, or hinged doors can be modeled as rigid bodies rotating about a single fixed axis. The ''z''-axis has been chosen by convention.

Position

This allows the description of a rotation as the angular position of a planar reference frame ''M'' relative to a fixed ''F'' about this shared ''z''-axis. Coordinates p = (''x'', ''y'') in ''M'' are related to coordinates P = (X, Y) in ''F'' by the matrix equation:
\mathbf(t) = (t)mathbf,

where
(t)= \begin
\cos(\theta(t)) & -\sin(\theta(t)) \\
\sin(\theta(t)) & \cos(\theta(t))
\end,
is the rotation matrix that defines the angular position of ''M'' relative to ''F'' as a function of time.

Velocity

If the point p does not move in ''M'', its velocity in ''F'' is given by
\mathbf_P = \dot = dot(t)mathbf.
It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(''t''),
\mathbf_P = dot(t)A(t)^]\mathbf = Omegamathbf,
where the matrix
$$$$Omega= \begin 0 & -\omega \\ \omega & 0 \end,
is known as the angular velocity matrix of ''M'' relative to ''F''. The parameter ''ω'' is the time derivative of the angle ''θ'', that is:
$\omega = \frac.$

Acceleration

The acceleration of P(''t'') in ''F'' is obtained as the time derivative of the velocity,
\mathbf_P = \ddot(t) = dotmathbf + Omegadot,
which becomes
\mathbf_P = dotmathbf + Omegadiv class="linkinfo_desc">Omega