# Kinematics

Kinematics is the branch of classical mechanics which describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without consideration of the causes of motion. Kinematics as a field of study is often referred to as the "geometry of motion". For further details, see Analytical dynamics.

To describe motion, kinematics studies the trajectories of points, lines, other geometric objects, and their differential properties such as velocity and acceleration. Kinematics is used in astrophysics to describe the motion of celestial bodies and systems, and in mechanical engineering, robotics, and biomechanics to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the skeleton of the human body.

The study of kinematics can be abstracted into purely mathematical functions. For instance, rotation can be represented by elements of the unit circle in the complex plane. Other planar algebras are used to represent the shear mapping of classical motion in absolute time and space and to represent the Lorentz transformations of relativistic space and time. By using time as a parameter in geometry, mathematicians have developed a science of kinematic geometry.

The use of geometric transformations, also called rigid transformations, to describe the movement of components of a mechanical system simplifies the derivation of its equations of motion, and is 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, kinematic synthesis designs a mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism.

## Etymology of the term

The term kinematic is the English version of A.M. Ampère's cinématique, which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν 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 but also the Greek word for writing. 

## Kinematics of a particle trajectory Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and and radial accelerations.
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.

Particle kinematics is the study of the properties of the trajectory of a particle. The position of a particle is defined to be 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 located at your home, such that East is the x-direction and North is the y-direction, then the coordinate vector to the base of the tower is r=(0, -50, 0). If the tower is 50 m high, then the coordinate vector to the top of the tower is r=(0, -50, 50).

Usually a three-dimensional coordinate systems is used to define the position of a particle. However, if the particle is constrained to lie in a plane or on a sphere, a two-dimensional coordinate system can be used. All observations in physics are incomplete without the reference frame being specified.

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 of point P can be expressed as $\mathbf{P} = (x_P,y_P,z_P) = x_P\hat{i} + y_P\hat{j} + z_P\hat{k},$

where xP, yP, and zP are the Cartesian coordinates and i, j and k are the unit vectors along the x, y, and z coordinate axes, respectively. The magnitude of the position vector |P| gives the distance between the point P and the origin. $|\mathbf{P}| = \sqrt{x_P^{\ 2} + y_P^{\ 2} + z_P^{\ 2}}.$

The direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the position vector of a particle isn't unique. The position vector of a given particle is different relative to different frames of reference.

The trajectory of a particle is a vector function of time, P(t), which defines the curve traced by the moving particle, given by $\mathbf{P}(t) = x_P(t)\hat{i} + y_P(t)\hat{j} +z_P(t) \hat{k},$

where the coordinates xP, yP, and zP are each functions of time.

### Velocity and speed

The velocity of a particle is a vector quantity that tells about the direction and magnitude of the rate of change of the position vector, that is, how the position of a point changes with each instant of time. Consider the ratio of the difference of two positions of a particle divided by the time interval, which is called the average velocity over that time interval. This average velocity is defined as $\overline{\mathbf{V}} = \frac {\Delta \mathbf{P}}{\Delta t} \ ,$

where ΔP is the difference in the position vector over the time interval Δt.

In the limit as the time interval Δt becomes smaller and smaller, the average velocity becomes the time derivative of the position vector, $\mathbf{V} = \lim_{\Delta t\rightarrow0}\frac{\Delta\mathbf{P}}{\Delta t} = \frac {d \mathbf{P}}{d t}=\dot{\mathbf{P}} = \dot{x}_p\hat{i}+\dot{y}_P\hat{j}+\dot{z}_P\hat{k}.$

Thus, velocity is the time rate of change of position, and the dot denotes the derivative with respect to time. Furthermore, the velocity is tangent to the trajectory of the particle.

As a position vector itself is frame dependent, therefore its velocity is also dependent on the reference frame.

The speed of an object is the magnitude |V| of its velocity. It is a scalar quantity: $|\mathbf{V}| = |\dot{\mathbf{P}} | = \frac {d s}{d t},$

where s is the arc-length measured along the trajectory of the particle. This arc-length traveled by a particle over time is a non-decreasing quantity. Hence, ds/dt is non-negative, which implies that speed is also non-negative.

### 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. $\overline{\mathbf{A}} = \frac {\Delta \mathbf{V}}{\Delta t} \ ,$

where ΔV is the difference in the velocity vector and Δt 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, $\mathbf{A} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{V}}{\Delta t} = \frac {d \mathbf{V}}{d t} = \dot{\mathbf{V}} = \ddot{\mathbf{P}} = \ddot{x}_p\hat{i}+\ddot{y}_P\hat{j}+\ddot{z}_P\hat{k}.$

Thus, acceleration is the second derivative of the position vector that defines the trajectory of a particle.

### Relative position vector

A relative position vector is a vector that defines the position of a particle relative to another particle. It is the difference in position of the two particles.

If point A has position PA = (xA,yA,zA) and point B has position PB = (xB,yB,zB), the displacement RB/A of B from A is given by: $\mathbf{R}_{B/A} = \mathbf{P}_B - \mathbf{P}_A = (x_B-x_A,y_B-y_A,z_B-z_A).$

Geometrically, the relative position vector RB/A is the vector from point A to point B. The values of the coordinate vectors of points vary with the choice of coordinate frame, however the relative position vector between a pair of points has the same length no matter what coordinate frame is used and is said to be frame invariant.

To describe the motion of a particle B relative to another particle A, we notice that the position B can be formulated as the position of A plus the position of B relative to A, that is: $\mathbf{P}_{B} = \mathbf{P}_{A} + (\mathbf{P}_{B} - \mathbf{P}_{A}) = \mathbf{P}_{A} + \mathbf{R}_{B/A}.$

### Relative velocity

The relations between relative positions vectors become relations between relative velocities by computing the time-derivative. The second time derivative yields relations for relative accelerations.

For example, let the particle B move with velocity VB and particle A move with velocity VA in a given reference frame. Then the velocity of B relative to A is given by: $\mathbf{V}_{B/A} = \mathbf{V}_{B} -\mathbf{V}_{A} \,\! .$

This can be obtained by computing the time derivative of the relative position vector RB/A.

This equation provides a formula for the velocity of B in terms of the velocity of A and its relative velocity, $\mathbf{V}_{B} = \mathbf{V}_{A} + \mathbf{V}_{B/A} \,\! .$

With a large velocity V, where the fraction V/c is significant, c being the speed of light, another scheme of relative velocity called rapidity, that depends on this ratio, is used in special relativity.

## Particle trajectories under constant acceleration

Newton's laws state that a constant force acting on a particle generates a constant acceleration. For example, a particle in a parallel gravity field experiences a force acting downwards that is proportional to the constant acceleration of gravity, and no force in the horizontal direction. This is called projectile motion.

If the acceleration vector A of a particle P is constant in magnitude and direction, the particle is said to be undergoing uniformly accelerated motion. In this case, the trajectory P(t) of the particle can be obtained by integrating the acceleration A with respect to time.

The first integral yields the velocity of the particle, $\mathbf{V}(t) = \int_0^{t} \mathbf{A} dt = \mathbf{A}t + \mathbf{V}_0.$

A second integration yields its trajectory, $\mathbf{P}(t) = \int_0^t \mathbf{V}(t) dt = \int(\mathbf{A}t + \mathbf{V}_0)dt = \tfrac{1}{2} \mathbf{A} t^2 + \mathbf{V}_0 t + \mathbf{P}_0.$

Additional relations between displacement, velocity, acceleration, and time can be derived. Since A = (VV0)/t, $\mathbf{P}(t) = \mathbf{P}_0 + \left(\frac{\mathbf{V}+ \mathbf{V}_0}{2}\right) t .$

By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.

A relationship without explicit time dependence may also be derived using the relation At = VV0, $(\mathbf{P} - \mathbf{P}_0) \cdot \mathbf{A} t = \left( \mathbf{V} - \mathbf{V}_0 \right) \cdot \frac{\mathbf{V} + \mathbf{V}_0}{2} t \ ,$

where · denotes the dot product. Divide both sides by t and expand the dot-products to obtain, $2(\mathbf{P} - \mathbf{P}_0) \cdot \mathbf{A} = |\mathbf{V}|^2 - |\mathbf{V}_0|^2.$

In the case of straight-line motion, where P and P0 are parallel to A, this equation becomes: $|\mathbf{V}|^2= |\mathbf{V}_0|^2 + 2 |\mathbf{A}|(|\mathbf{P}-\mathbf{P}_0|).$

This can be simplified using the notation |A|=a, |V|=v, and |P|=r, so $v^2= v_0^2 + 2a(r-r_0).$

This relation is useful when time is not known explicitly. Figure 2: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component aθ that increases the rate of rotation: dω/dt = |aθ|/R.

## Particle trajectories in cylindrical-polar coordinates

It is often convenient to formulate the trajectory of a particle P(t) = (X(t), Y(t) and 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 P measured in a fixed reference frame F. As the particle moves, its coordinate vector P(t) traces its trajectory, which is a curve in space, given by: $\textbf{P}(t) = X(t)\hat{i} + Y(t)\hat{j} + Z(t)\hat{k},$

where i, j, and k are the unit vectors along the X, Y and Z axes of the reference frame F, respectively.

Consider a particle P that moves on the surface of a circular cylinder, 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, $\textbf{P}(t) = R\cos\theta(t)\hat{i} + R\sin\theta(t)\hat{j} + Z(t)\hat{k}.$

The cylindrical coordinates for P(t) can be simplified by introducing the radial and tangential unit vectors, $\textbf{e}_r = \cos\theta(t)\hat{i} + \sin\theta(t)\hat{j}, \quad \textbf{e}_t = -\sin\theta(t)\hat{i} + \cos\theta(t)\hat{j}.$

Using this notation, P(t) takes the form, $\textbf{P}(t) = R\textbf{e}_r + Z(t)\hat{k},$

where R is constant.

In general, the trajectory P(t) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory in cylindrical-polar coordinates becomes: $\textbf{P}(t) = R(t)\textbf{e}_r + Z(t)\hat{k}.$

The velocity vector VP is the time derivative of the trajectory P(t), which yields: $\textbf{V}_P = \frac{d}{dt}(R(t)\textbf{e}_r + Z(t)\hat{k}) = \dot{R}\textbf{e}_r + R\dot{\theta}\textbf{e}_t + \dot{Z}\hat{k},$

where $\frac{d}{dt}\textbf{e}_r = \dot{\theta}\textbf{e}_t.$

In this case, the acceleration AP, which is the time derivative of the velocity VP, is given by: $\textbf{A}_P = \frac{d}{dt}(\dot{R}\textbf{e}_r + R\dot{\theta}\textbf{e}_t + \dot{Z}(t)\hat{k}) = (\ddot{R} - R\dot{\theta}^2)\textbf{e}_r + (R\ddot{\theta} + 2\dot{R}\dot{\theta})\textbf{e}_t + \ddot{Z}(t)\hat{k}.$

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 VP is the time derivative of the trajectory P(t), $\textbf{V}_P = \frac{d}{dt}(R\textbf{e}_r + Z(t)\hat{k}) = R\dot{\theta}\textbf{e}_t + \dot{Z}\hat{k}.$

The acceleration vector becomes: $\textbf{A}_P = \frac{d}{dt}(R\dot{\theta}\textbf{e}_t + \dot{Z}\hat{k}) = - R\dot{\theta}^2\textbf{e}_r + R\ddot{\theta}\textbf{e}_t + \ddot{Z}\hat{k}.$

### Planar circular trajectories

A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the Z axis: $\textbf{P}(t) = R\textbf{e}_r + Z_0\hat{k},$

where R and Z0 are constants. In this case, the velocity VP is given by: $\textbf{V}_P = \frac{d}{dt}(R\textbf{e}_r + Z_0\hat{k}) = R\dot{\theta}\textbf{e}_t =R\omega\textbf{e}_t,$

where $\omega = \dot{\theta},$

is the angular velocity of the unit vector et around the z axis of the cylinder.

The acceleration AP of the particle P is now given by: $\textbf{A}_P = \frac{d}{dt}(R\dot{\theta}\textbf{e}_t) = - R\dot{\theta}^2\textbf{e}_r + R\ddot{\theta}\textbf{e}_t.$

The components $a_r = - R\dot{\theta}^2, \quad a_t = R\ddot{\theta},$

are called, respectively, the radial and tangential components of acceleration.

The notation for angular velocity and angular acceleration is often defined as $\omega = \dot{\theta}, \quad \alpha = \ddot{\theta},$

so the radial and tangential acceleration components for circular trajectories are also written as $a_r = - R\omega^2, \quad a_t = R\alpha.$

## Point trajectories in a body moving in the plane

The movement of components of a mechanical system is analyzed by attaching a reference frame to each part and determining how the reference frames move relative to each other. If the structural strength of the parts are sufficient then their deformation can be neglected and rigid transformations used to define this relative movement. This brings geometry into the study of mechanical movement.

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. Perhaps best known is high-school Euclidean geometry where planar triangles are studied under congruent transformations (also called isometries or rigid transformations). These transformations displace the triangle in the plane without changing the angle at each vertex or the distances between vertices. 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 the plane are two-dimensional vectors in R2, so rigid transformations are those that preserve the distance measured between any two points. The Euclidean distance formula is simply the Pythagorean theorem. The set of rigid transformations in an n-dimensional space is called the special Euclidean group on Rn, and denoted SE(n).

### Displacements and motion The movement of each of the components of the Boulton & Watt Steam Engine (1784) is modeled by a continuous set of rigid displacements.

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 3x3 matrix known as a homogeneous transform. The 3x3 homogeneous transform is constructed from a 2x2 rotation matrix A(φ) and the 2x1 translation vector d=(dx, dy), as: $[T(\phi, \mathbf{d})] = \begin{bmatrix} A(\phi) & \mathbf{d} \\ 0, 0 & 1\end{bmatrix} = \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix}.$

These homogeneous transforms perform rigid transformations on the points in the plane z=1, that is on points with coordinates p=(x, y, 1).

In particular, let p 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: $\textbf{P} = [T(\phi, \mathbf{d})]\textbf{p} = \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix}\begin{Bmatrix}x\\y\\1\end{Bmatrix}.$

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 to be 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 relative to the fixed frame F, the motion is said to be 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: $\textbf{P}(t)=[T(0,\textbf{d}(t))]\textbf{p} = \textbf{d}(t) + \textbf{p}.$

Thus, for bodies in pure translation, the velocity and acceleration of every point P in the body are given by: $\textbf{V}_P=\dot{\textbf{P}}(t) = \dot{\textbf{d}}(t)=\textbf{V}_O,\quad \textbf{A}_P=\ddot{\textbf{P}}(t) = \ddot{\textbf{d}}(t) = \textbf{A}_O,$

where the dot denotes the derivative with respect to time and VO and AO 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 Figure 1: The angular velocity vector Ω points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule. Angular position θ(t) changes with time at a rate ω(t) = dθ/dt.

Rotational or angular kinematics is the description of the rotation of an object. The description of rotation requires some method for describing orientation. Common descriptions include Euler angles and the kinematics of turns induced by algebraic products.

In what follows, attention is restricted to simple rotation about an axis of fixed orientation. The z-axis has been chosen for convenience.

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: $\mathbf{P}(t) = [A(t)]\mathbf{p},$
where $[A(t)] = \begin{bmatrix} \cos\theta(t) & -\sin\theta(t) \\ \sin\theta(t) & \cos\theta(t) \end{bmatrix},$
is the rotation matrix that defines the angular position of M relative to F.
Velocity
If the point p does not move in M, its velocity in F is given by $\mathbf{V}_P = \dot{\mathbf{P}} = [\dot{A}(t)]\mathbf{p}.$
It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(t), $\mathbf{V}_P = [\dot{A}(t)][A(t)^{-1}]\mathbf{P} = [\Omega]\mathbf{P},$
where the matrix $[\Omega] = \begin{bmatrix} 0 & -\omega \\ \omega & 0 \end{bmatrix},$
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: $\omega = \frac{d\theta}{dt}.$
Acceleration
The acceleration of P(t) in F is obtained as the time derivative of the velocity, $\mathbf{A}_P = \ddot{P}(t) = [\dot{\Omega}]\mathbf{P} + [\Omega]\dot{\mathbf{P}},$
which becomes $\mathbf{A}_P = [\dot{\Omega}]\mathbf{P} + [\Omega][\Omega]\mathbf{P},$
where $[\dot{\Omega}] = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix},$
is the angular acceleration matrix of M on F, and $\alpha = \frac{d^2\theta}{dt^2}.$

The description of rotation then involves these three quantities:

• Angular position : the oriented distance from a selected origin on the rotational axis to a point of an object is a vector r ( t ) locating the point. The vector r(t) has some projection (or, equivalently, some component) r(t) on a plane perpendicular to the axis of rotation. Then the angular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r(t) in a known rotation sense (typically given by the right-hand rule).
• Angular velocity : the angular velocity ω is the rate at which the angular position θ changes with respect to time t: $\omega = \frac {d\theta}{dt}$
The angular velocity is represented in Figure 1 by a vector Ω pointing along the axis of rotation with magnitude ω and sense determined by the direction of rotation as given by the right-hand rule.
• Angular acceleration : the magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t: $\alpha = \frac {d\omega}{dt}$

The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges: $\omega_{\mathrm{f}} = \omega_{\mathrm{i}} + \alpha t\!$ $\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \omega_{\mathrm{i}} t + \tfrac{1}{2} \alpha t^2$ $\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \tfrac{1}{2} (\omega_{\mathrm{f}} + \omega_{\mathrm{i}})t$ $\omega_{\mathrm{f}}^2 = \omega_{\mathrm{i}}^2 + 2 \alpha (\theta_{\mathrm{f}} - \theta_{\mathrm{i}}).$

Here θi and θf are, respectively, the initial and final angular positions, ωi and ωf are, respectively, the initial and final angular velocities, and α is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.

## Point trajectories in body moving in three dimensions

Important formulas in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either Newton's second law or Lagrange's equations.

### Position

In order to define these formulas, the movement of a component B of a mechanical system is defined by the set of rotations [A(t)] and translations d(t) assembled into the homogeneous transformation [T(t)]=[A(t), d(t)]. If p is the coordinates of a point P in B measured in the moving reference frame M, then the trajectory of this point traced in F is given by: $\textbf{P}(t)=[T(t)]\textbf{p} = \begin{Bmatrix} \textbf{P} \\ 1\end{Bmatrix}=\begin{bmatrix} A(t) & \textbf{d}(t) \\ 0 & 1\end{bmatrix} \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.$

This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context.

This equation for the trajectory of P can be inverted to compute the coordinate vector p in M as: $\textbf{p} = [T(t)]^{-1}\textbf{P}(t) = \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}=\begin{bmatrix} A(t)^T & -A(t)^T\textbf{d}(t) \\ 0 & 1\end{bmatrix} \begin{Bmatrix} \textbf{P}(t) \\ 1\end{Bmatrix}.$

This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is: $[A(t)]^T[A(t)]=I.\!$

### Velocity

The velocity of the point P along its trajectory P(t) is obtained as the time derivative of this position vector, $\textbf{V}_P = [\dot{T}(t)]\textbf{p} = \begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}(t) & \dot{\textbf{d}}(t) \\ 0 & 0 \end{bmatrix} \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.$

The dot denotes the derivative with respect to time; because p is constant, its derivative is zero.

This formula can be modified to obtain the velocity of P by operating on its trajectory P(t) measured in the fixed frame F. Substituting the inverse transform for p into the velocity equation yields: $\textbf{V}_P = [\dot{T}(t)][T(t)]^{-1}\textbf{P}(t) = \begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}A^T & -\dot{A}A^T\textbf{d} + \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix} \begin{Bmatrix} \textbf{P}(t) \\ 1\end{Bmatrix}=[S]\textbf{P}.$

The matrix [S] is given by: $[S] = \begin{bmatrix} \Omega & -\Omega\textbf{d} + \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix}$

where $[\Omega] = \dot{A}A^T,$

is the angular velocity matrix.

Multiplying by the operator [S], the formula for the velocity VP takes the form: $\textbf{V}_P = [\Omega](\textbf{P}-\textbf{d}) + \dot{\textbf{d}} = \omega\times \textbf{R}_{P/O} + \textbf{V}_O,$

where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω]; the vector $\textbf{R}_{P/O}=\textbf{P}-\textbf{d},$

is the position of P relative to the origin O of the moving frame M; and $\textbf{V}_O=\dot{\textbf{d}},$

is the velocity of the origin O.

### Acceleration

The acceleration of a point P in a moving body B is obtained as the time derivative of its velocity vector: $\textbf{A}_P = \frac{d}{dt}\textbf{V}_P = \frac{d}{dt}\big([S]\textbf{P}\big)=[\dot{S}]\textbf{P} + [S]\dot{\textbf{P}} = [\dot{S}]\textbf{P} + [S][S]\textbf{P} .$

This equation can be expanded firstly by computing $[\dot{S}] = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\textbf{d} -\Omega\dot{\textbf{d}} + \ddot{\textbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\textbf{d} -\Omega\textbf{V}_O + \textbf{A}_O \\ 0 & 0 \end{bmatrix}$

and $[S]^2 = \begin{bmatrix} \Omega & -\Omega\textbf{d} + \textbf{V}_O \\ 0 & 0 \end{bmatrix}^2 = \begin{bmatrix} \Omega^2 & -\Omega^2\textbf{d} + \Omega\textbf{V}_O \\ 0 & 0 \end{bmatrix}.$

The formula for the acceleration AP can now be obtained as: $\textbf{A}_P = \dot{\Omega}(\textbf{P} - \textbf{d}) + \textbf{A}_O + \Omega^2(\textbf{P}-\textbf{d}),$

or $\textbf{A}_P = \alpha\times\textbf{R}_{P/O} + \omega\times\omega\times\textbf{R}_{P/O} + \textbf{A}_O,$

where α is the angular acceleration vector obtained from the derivative of the angular velocity matrix; $\textbf{R}_{P/O}=\textbf{P}-\textbf{d},$

is the relative position vector; and $\textbf{A}_O = \ddot{\textbf{d}}$

is the acceleration of the origin of the moving frame M.

## Kinematic constraints

Kinematic constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called holonomic constraints, and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called non-holonomic constraints. The following are some common examples.

### Kinematic coupling

A kinematic coupling exactly constrains all 6 degrees of freedom.

### Rolling without slipping

An object that rolls against a surface without slipping obeys the condition that the velocity of its center of mass is equal to the cross product of its angular velocity with a vector from the point of contact to the center of mass: $\boldsymbol{ v}_G(t) = \boldsymbol{\Omega} \times \boldsymbol{ r}_{G/O}.$

For the case of an object that does not tip or turn, this reduces to $v = r \omega$.

### Inextensible cord

This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero. A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. An equilibrium problem (i.e. not kinematic) of this type is the catenary.

### Kinematic pairs

Reuleaux called the ideal connections between components that form a machine kinematic pairs. He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification.

#### Lower pair

A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases:

• A revolute pair, or hinged joint, requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom, which is pure rotation about the axis of the hinge.
• A prismatic joint, or slider, requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body maintain contact with a similar parallel plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom. This degree of freedom is the distance of the slide along the line.
• A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body. It is a combination of a revolute joint and a sliding joint. This joint has two degrees of freedom. The position of the moving body is defined by both the rotation about and slide along the axis.
• A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom.
• A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.

#### Higher pairs

Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a cam joint. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.

### Kinematic chains

Rigid bodies ("links") connected by kinematic pairs ("joints") are known as kinematic chains. Mechanisms and robots are examples of kinematic chains. The degree of freedom of a kinematic chain is computed from the number of links and the number and type of joints using the mobility formula. This formula can also be used to enumerate the topologies of kinematic chains that have a given degree of freedom, which is known as type synthesis in machine design.

#### Examples

The planar one degree-of-freedom linkages assembled from N links and j hinged or sliding joints are:

• N=2, j=1 : a two-bar linkage that is the lever;
• N=4, j=4 : the four-bar linkage;
• N=6, j=7 : a six-bar linkage. This must have two links ("ternary links") that support three joints. There are two distinct topologies that depend on how the two ternary linkages are connected. In the Watt topology, the two ternary links have a common joint; in the Stephenson topology, the two ternary links do not have a common joint and are connected by binary links.
• N=8, j=10 : eight-bar linkage with 16 different topologies;
• N=10, j=13 : ten-bar linkage with 230 different topologies;
• N=12, j=16 : twelve-bar linkage with 6,856 topologies.

For larger chains and their linkage topologies, see R. P. Sunkari and L. C. Schmidt, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", Mechanism and Machine Theory #41, pp. 1021–1030 (2006).