Kinematics

16 min readDec 4, 2023

Kinematics is a subfield of physics, 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, as a field of study, is often referred to as the “geometry of motion” and is occasionally seen as a branch of mathematics. 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. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

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, also 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. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism.

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).

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

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 r can be expressed as

where x x, y y, and z z are the Cartesian coordinates and x ^ are the unit vectors along the x x, y y, and z z coordinate axes, respectively. The magnitude of the position vector | r | gives the distance between the point r and the origin.

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, r ( t ), which defines the curve traced by the moving particle, given by

where x ( t ) x(t), y ( t ) y(t), and z ( t ) 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

where Δ r is the displacement vector during the time interval Δ t. In the limit that the time interval Δ t approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector,

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:

where s s is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, d s / d t 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.

where Δv is the average velocity 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,

Alternatively,

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:

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

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

If point A has position components r A = ( x A , y A , z A )

and point B has position components r B = ( x B , y B , z B )

then the position of point A relative to point B is the difference between their components:

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:

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,

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,

and their time derivatives from elementary calculus:

Using this notation, r(t) takes the form,

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:

Where r, θ, and z might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector vP is the time derivative of the trajectory r(t), which yields:

Similarly, the acceleration aP, which is the time derivative of the velocity vP, is given by:

The term

acts toward the center of curvature of the path at that point on the path, is commonly called the centripetal acceleration. The term

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 vP is the time derivative of the trajectory r(t),

Planar circular trajectories

A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the z axis:

where r and z0 are constants. In this case, the velocity vP is given by:

where ω is the angular velocity of the unit vector θ^ around the z axis of the cylinder.

The acceleration aP of the particle P is now given by:

The components

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

The notation for angular velocity and angular acceleration is often defined as

so the radial and tangential acceleration components for circular trajectories are also written as

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.[20] 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 Rn, 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 = (dx, dy), as:

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:

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.

Rotation of a body around a fixed axis

Rotational or angular kinematics is the description of the rotation of an object. 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:

where

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

It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(t),

where the matrix

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:

Acceleration

The acceleration of P(t) in F is obtained as the time derivative of the velocity,

which becomes

where

is the angular acceleration matrix of M on F, and

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:
ω = d θ d t

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:

The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:

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:

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:

This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is:

Velocity

The velocity of the point P along its trajectory P(t) is obtained as the time derivative of this position vector,

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: