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.^{[1]}^{[2]}^{[3]} The term is the English version of A.M. Ampère's cinématique,^{[4]}
which he constructed from the Greek κίνημα, kinema (movement, motion), derived from κινεῖν, kinein (to move).^{[5]}
^{[6]}
The study of kinematics is often referred to as the geometry of motion.^{[7]} (See analytical dynamics for more detail on usage.)
To describe motion, kinematics studies the trajectories of points, lines and 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^{[8]} 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 expressions. 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.^{[9]} In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system, or mechanism.
Kinematics of a particle trajectory
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
- $\backslash mathbf\{P\}\; =\; (x\_P,y\_P,z\_P)\; =\; x\_P\backslash vec\{i\}\; +\; y\_P\backslash vec\{j\}\; +\; z\_P\backslash vec\{k\},$
where x_{P}, y_{P}, and z_{P} 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.
- $|\backslash mathbf\{P\}|\; =\; \backslash sqrt\{x\_P^\{\backslash \; 2\}\; +\; y\_P^\{\backslash \; 2\}\; +\; z\_P^\{\backslash \; 2\}\}.$
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
- $\backslash mathbf\{P\}(t)\; =\; x\_P(t)\backslash vec\{i\}\; +\; y\_P(t)\backslash vec\{j\}\; +z\_P(t)\; \backslash vec\{k\},$
where the coordinates x_{P}, y_{P}, and z_{P} are each functions of time.
Velocity and speed
The velocity of a particle is a vector 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
- $\backslash overline\{\backslash mathbf\{V\}\}\; =\; \backslash frac\; \{\backslash Delta\; \backslash mathbf\{P\}\}\{\backslash Delta\; t\}\; \backslash \; ,$
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,
- $\backslash mathbf\{V\}\; =\; \backslash lim\_\{\backslash Delta\; t\backslash rightarrow0\}\backslash frac\{\backslash Delta\backslash mathbf\{P\}\}\{\backslash Delta\; t\}\; =\; \backslash frac\; \{d\; \backslash mathbf\{P\}\}\{d\; t\}=\backslash dot\{\backslash mathbf\{P\}\}\; =\; \backslash dot\{x\}\_p\backslash vec\{i\}+\backslash dot\{y\}\_P\backslash vec\{j\}+\backslash dot\{z\}\_P\backslash vec\{k\}.$
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:
- $|\backslash mathbf\{V\}|\; =\; |\backslash dot\{\backslash mathbf\{P\}\}\; |\; =\; \backslash frac\; \{d\; s\}\{d\; t\},$
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
- $\backslash overline\{\backslash mathbf\{A\}\}\; =\; \backslash frac\; \{\backslash Delta\; \backslash mathbf\{V\}\}\{\backslash Delta\; t\}\; \backslash \; ,$
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,
- $\backslash mathbf\{A\}\; =\; \backslash lim\_\{\backslash Delta\; t\; \backslash rightarrow\; 0\}\; \backslash frac\{\backslash Delta\; \backslash mathbf\{V\}\}\{\backslash Delta\; t\}\; =\; \backslash frac\; \{d\; \backslash mathbf\{V\}\}\{d\; t\}\; =\; \backslash dot\{\backslash mathbf\{V\}\}\; =\; \backslash ddot\{\backslash mathbf\{P\}\}\; =\; \backslash ddot\{x\}\_p\backslash vec\{i\}+\backslash ddot\{y\}\_P\backslash vec\{j\}+\backslash ddot\{z\}\_P\backslash vec\{k\}.$
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 P_{A} = (x_{A},y_{A},z_{A}) and point B has position P_{B} = (x_{B},y_{B},z_{B}), the displacement R_{B/A} of B from A is given by
- $\backslash mathbf\{R\}\_\{B/A\}\; =\; \backslash mathbf\{P\}\_B\; -\; \backslash mathbf\{P\}\_A\; =\; (x\_B-x\_A,y\_B-y\_A,z\_B-z\_A).$
Geometrically, the relative position vector R_{B/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
- $\backslash mathbf\{P\}\_\{B\}\; =\; \backslash mathbf\{P\}\_\{A\}\; +\; (\backslash mathbf\{P\}\_\{B\}\; -\; \backslash mathbf\{P\}\_\{A\})\; =\; \backslash mathbf\{P\}\_\{A\}\; +\; \backslash mathbf\{R\}\_\{B/A\}.$
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 V_{B} and particle A move with velocity V_{A} in a given reference frame. Then the velocity of B relative to A is given by
- $\backslash mathbf\{V\}\_\{B/A\}\; =\; \backslash mathbf\{V\}\_\{B\}\; -\backslash mathbf\{V\}\_\{A\}\; \backslash ,\backslash !\; .$
This can be obtained by computing the time derivative of the relative position vector R_{B/A}.
This equation provides a formula for the velocity of B in terms of the velocity of A and its relative velocity,
- $\backslash mathbf\{V\}\_\{B\}\; =\; \backslash mathbf\{V\}\_\{A\}\; +\; \backslash mathbf\{V\}\_\{B/A\}\; \backslash ,\backslash !\; .$
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,
- $\backslash mathbf\{V\}(t)\; =\; \backslash int\_0^\{t\}\; \backslash mathbf\{A\}\; dt\; =\; \backslash mathbf\{A\}t\; +\; \backslash mathbf\{V\}\_0.$
A second integration yields its trajectory,
- $\backslash mathbf\{P\}(t)\; =\; \backslash int\_0^t\; \backslash mathbf\{V\}(t)\; dt\; =\; \backslash int(\backslash mathbf\{A\}t\; +\; \backslash mathbf\{V\}\_0)dt\; =\; \backslash tfrac\{1\}\{2\}\; \backslash mathbf\{A\}\; t^2\; +\; \backslash mathbf\{V\}\_0\; t\; +\; \backslash mathbf\{P\}\_0.$
Additional relations between displacement, velocity, acceleration, and time can be derived. Since A = (V − V_{0})/t,
- $\backslash mathbf\{P\}(t)\; =\; \backslash mathbf\{P\}\_0\; +\; \backslash left(\backslash frac\{\backslash mathbf\{V\}+\; \backslash mathbf\{V\}\_0\}\{2\}\backslash right)\; t\; .$
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 = V − V_{0},
- $(\backslash mathbf\{P\}\; -\; \backslash mathbf\{P\}\_0)\; \backslash cdot\; \backslash mathbf\{A\}\; t\; =\; \backslash left(\; \backslash mathbf\{V\}\; -\; \backslash mathbf\{V\}\_0\; \backslash right)\; \backslash cdot\; \backslash frac\{\backslash mathbf\{V\}\; +\; \backslash mathbf\{V\}\_0\}\{2\}\; t\; \backslash \; ,$
where · denotes the dot product. Divide both sides by t and expand the dot-products to obtain,
- $2(\backslash mathbf\{P\}\; -\; \backslash mathbf\{P\}\_0)\; \backslash cdot\; \backslash mathbf\{A\}\; =\; |\backslash mathbf\{V\}|^2\; -\; |\backslash mathbf\{V\}\_0|^2.$
In the case of straight-line motion, where P and P_{0} are parallel to A, this equation becomes
- $|\backslash mathbf\{V\}|^2=\; |\backslash mathbf\{V\}\_0|^2\; +\; 2\; |\backslash mathbf\{A\}|(|\backslash mathbf\{P\}-\backslash mathbf\{P\}\_0|).$
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.
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
- $\backslash textbf\{P\}(t)\; =\; X(t)\backslash vec\{i\}\; +\; Y(t)\backslash vec\{j\}\; +\; Z(t)\backslash vec\{k\},$
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,
- $\backslash textbf\{P\}(t)\; =\; R\backslash cos\backslash theta(t)\backslash vec\{i\}\; +\; R\backslash sin\backslash theta(t)\backslash vec\{j\}\; +\; Z(t)\backslash vec\{k\}.$
The cylindrical coordinates for P(t) can be simplified by introducing the radial and tangential unit vectors,
- $\backslash textbf\{e\}\_r\; =\; \backslash cos\backslash theta(t)\backslash vec\{i\}\; +\; \backslash sin\backslash theta(t)\backslash vec\{j\},\; \backslash quad\; \backslash textbf\{e\}\_t\; =\; -\backslash sin\backslash theta(t)\backslash vec\{i\}\; +\; \backslash cos\backslash theta(t)\backslash vec\{j\}.$
Using this notation, P(t) takes the form,
- $\backslash textbf\{P\}(t)\; =\; R\backslash textbf\{e\}\_r\; +\; Z(t)\backslash vec\{k\},$
where R is constant.
Now, 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-polor coordinates becomes
- $\backslash textbf\{P\}(t)\; =\; R(t)\backslash textbf\{e\}\_r\; +\; Z(t)\backslash vec\{k\}.$
The velocity vector V_{P} is the time derivative of the trajectory P(t), which yields,
- $\backslash textbf\{V\}\_P\; =\; \backslash frac\{d\}\{dt\}(R(t)\backslash textbf\{e\}\_r\; +\; Z(t)\backslash vec\{k\})\; =\; \backslash dot\{R\}\backslash textbf\{e\}\_r\; +\; R\backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t\; +\; \backslash dot\{Z\}\backslash vec\{k\},$
where
- $\backslash frac\{d\}\{dt\}\backslash textbf\{e\}\_r\; =\; \backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t.$
In this case, the acceleration A_{P}, which is the time derivative of the velocity V_{P}, is given by
- $\backslash textbf\{A\}\_P\; =\; \backslash frac\{d\}\{dt\}(\backslash dot\{R\}\backslash textbf\{e\}\_r\; +\; R\backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t\; +\; \backslash dot\{Z\}(t)\backslash vec\{k\})\; =\; (\backslash ddot\{R\}\; -\; R\backslash dot\{\backslash theta\}^2)\backslash textbf\{e\}\_r\; +\; (R\backslash ddot\{\backslash theta\}\; +\; 2\backslash dot\{R\}\backslash dot\{\backslash theta\})\backslash textbf\{e\}\_t\; +\; \backslash ddot\{Z\}(t)\backslash vec\{k\}.$
If the radius is constant
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 P(t),
- $\backslash textbf\{V\}\_P\; =\; \backslash frac\{d\}\{dt\}(R\backslash textbf\{e\}\_r\; +\; Z(t)\backslash vec\{k\})\; =\; R\backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t\; +\; \backslash dot\{Z\}\backslash vec\{k\}.$
The acceleration vector becomes
- $\backslash textbf\{A\}\_P\; =\; \backslash frac\{d\}\{dt\}(R\backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t\; +\; \backslash dot\{Z\}\backslash vec\{k\})\; =\; -\; R\backslash dot\{\backslash theta\}^2\backslash textbf\{e\}\_r\; +\; R\backslash ddot\{\backslash theta\}\backslash textbf\{e\}\_t\; +\; \backslash ddot\{Z\}\backslash vec\{k\}.$
Planar circular trajectories
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the Z axis, in which case
- $\backslash textbf\{P\}(t)\; =\; R\backslash textbf\{e\}\_r\; +\; Z\_0\backslash vec\{k\},$
where R and Z_{0} are constants. In this case, the velocity V_{P} is given by
- $\backslash textbf\{V\}\_P\; =\; \backslash frac\{d\}\{dt\}(R\backslash textbf\{e\}\_r\; +\; Z\_0\backslash vec\{k\})\; =\; R\backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t\; =R\backslash omega\backslash textbf\{e\}\_t,$
where
- $\backslash omega\; =\; \backslash dot\{\backslash theta\},$
is the angular velocity of the unit vector e_{t} around the z axis of the cylinder.
The acceleration A_{P} of the particle P is now given by
- $\backslash textbf\{A\}\_P\; =\; \backslash frac\{d\}\{dt\}(R\backslash dot\{\backslash theta\}\backslash textbf\{e\}\_t)\; =\; -\; R\backslash dot\{\backslash theta\}^2\backslash textbf\{e\}\_r\; +\; R\backslash ddot\{\backslash theta\}\backslash textbf\{e\}\_t.$
The components
- $a\_r\; =\; -\; R\backslash dot\{\backslash theta\}^2,\; \backslash quad\; a\_t\; =\; R\backslash ddot\{\backslash theta\},$
are called the radial and tangential components of acceleration, respectively.
The notation for angular velocity and angular acceleration is often defined as
- $\backslash omega\; =\; \backslash dot\{\backslash theta\},\; \backslash quad\; \backslash alpha\; =\; \backslash ddot\{\backslash theta\},$
so the radial and tangential acceleration components for circular trajectories are also written as
- $a\_r\; =\; -\; R\backslash omega^2,\; \backslash quad\; a\_t\; =\; R\backslash alpha.$
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.^{[11]}
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 R^{2}, 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 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 R^{2} can be represented by a certain type of 3x3 matrix known as a homogeneous transform. The 3x3 homogenous transform is constructed from a 2x2 rotation matrix A(φ) and the 2x1 translation vector d=(d_{x}, d_{y}), as
- $[T(\backslash phi,\; \backslash mathbf\{d\})]\; =\; \backslash begin\{bmatrix\}\; A(\backslash phi)\; \&\; \backslash mathbf\{d\}\; \backslash \backslash \; 0,\; 0\; \&\; 1\backslash end\{bmatrix\}$
= \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
- $\backslash textbf\{P\}\; =\; [T(\backslash phi,\; \backslash mathbf\{d\})]\backslash textbf\{p\}\; =\; \backslash begin\{bmatrix\}\; \backslash cos\backslash phi\; \&\; -\backslash sin\backslash phi\; \&\; d\_x\; \backslash \backslash \; \backslash sin\backslash phi\; \&\; \backslash cos\backslash phi\; \&\; d\_y\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\backslash end\{bmatrix\}\backslash begin\{Bmatrix\}x\backslash \backslash y\backslash \backslash 1\backslash end\{Bmatrix\}.$
Homogeneous transforms represent affine transformations. This formulation is necessary because a translation is not a linear transformation of R^{2}. However, using projective geometry, so that R^{2} is considered to be a subset of R^{3}, translations become affine linear transformations.^{[12]}
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,
- $\backslash textbf\{P\}(t)=[T(0,\backslash textbf\{d\}(t))]\backslash textbf\{p\}\; =\; \backslash textbf\{d\}(t)\; +\; \backslash textbf\{p\}.$
Thus, for bodies in pure translation the velocity and acceleration of every point P in the body are given by
- $\backslash textbf\{V\}\_P=\backslash dot\{\backslash textbf\{P\}\}(t)\; =\; \backslash dot\{\backslash textbf\{d\}\}(t)=\backslash textbf\{V\}\_O,\backslash quad\; \backslash textbf\{A\}\_P=\backslash ddot\{\backslash textbf\{P\}\}(t)\; =\; \backslash ddot\{\backslash textbf\{d\}\}(t)\; =\; \backslash textbf\{A\}\_O,$
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
Rotational or angular kinematics is the description of the rotation of an object.^{[13]} 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:
- $\backslash mathbf\{P\}(t)\; =\; [A(t)]\backslash mathbf\{p\},$
where
- $[A(t)]\; =\; \backslash begin\{bmatrix\}$
\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, then its velocity in F is given by
- $\backslash mathbf\{V\}\_P\; =\; \backslash dot\{\backslash mathbf\{P\}\}\; =\; [\backslash dot\{A\}(t)]\backslash mathbf\{p\}.$
It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(t),
- $\backslash mathbf\{V\}\_P\; =\; [\backslash dot\{A\}(t)][A(t)^\{-1\}]\backslash mathbf\{P\}\; =\; [\backslash Omega]\backslash mathbf\{P\},$
where the matrix
- $[\backslash Omega]\; =$
\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
- $\backslash omega\; =\; \backslash frac\{d\backslash theta\}\{dt\}.$
Acceleration: The acceleration of P(t) in F is obtained as the time derivative of the velocity,
- $\backslash mathbf\{A\}\_P\; =\; \backslash ddot\{P\}(t)\; =\; [\backslash dot\{\backslash Omega\}]\backslash mathbf\{P\}\; +\; [\backslash Omega]\backslash dot\{\backslash mathbf\{P\}\},$
which becomes
- $\backslash mathbf\{A\}\_P\; =\; [\backslash dot\{\backslash Omega\}]\backslash mathbf\{P\}\; +\; [\backslash Omega][\backslash Omega]\backslash mathbf\{P\},$
where
- $[\backslash dot\{\backslash Omega\}]\; =$
\begin{bmatrix}
0 & -\alpha \\ \alpha & 0
\end{bmatrix},
is the angular acceleration matrix of M on F, and
- $\backslash alpha\; =\; \backslash frac\{d^2\backslash theta\}\{dt^2\}.$
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:
- $\backslash omega\; =\; \backslash frac\; \{d\backslash theta\}\{dt\}$
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:
- $\backslash alpha\; =\; \backslash frac\; \{d\backslash omega\}\{dt\}$
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:
- $\backslash omega\_\{\backslash mathrm\{f\}\}\; =\; \backslash omega\_\{\backslash mathrm\{i\}\}\; +\; \backslash alpha\; t\backslash !$
- $\backslash theta\_\{\backslash mathrm\{f\}\}\; -\; \backslash theta\_\{\backslash mathrm\{i\}\}\; =\; \backslash omega\_\{\backslash mathrm\{i\}\}\; t\; +\; \backslash tfrac\{1\}\{2\}\; \backslash alpha\; t^2$
- $\backslash theta\_\{\backslash mathrm\{f\}\}\; -\; \backslash theta\_\{\backslash mathrm\{i\}\}\; =\; \backslash tfrac\{1\}\{2\}\; (\backslash omega\_\{\backslash mathrm\{f\}\}\; +\; \backslash omega\_\{\backslash mathrm\{i\}\})t$
- $\backslash omega\_\{\backslash mathrm\{f\}\}^2\; =\; \backslash omega\_\{\backslash mathrm\{i\}\}^2\; +\; 2\; \backslash alpha\; (\backslash theta\_\{\backslash mathrm\{f\}\}\; -\; \backslash theta\_\{\backslash mathrm\{i\}\}).$
Here θ_{i} and θ_{f} are, respectively, the initial and final angular positions, ω_{i} and ω_{f} 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 homogenous transformation [T(t)]=[A(t), d(t)]. Let p be 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
- $\backslash textbf\{P\}(t)=[T(t)]\backslash textbf\{p\}\; =$
\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,
- $\backslash textbf\{p\}\; =\; [T(t)]^\{-1\}\backslash textbf\{P\}(t)\; =$
\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
- $[A(t)]^T[A(t)]=I.\backslash !$
Velocity
The velocity of the point P along its trajectory P(t) is obtained as the time derivative of this position vector,
- $\backslash textbf\{V\}\_P\; =\; [\backslash dot\{T\}(t)]\backslash textbf\{p\}\; =$
\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, and 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. Substitute the inverse transform for p into the velocity equation to obtain
- $\backslash textbf\{V\}\_P\; =\; [\backslash dot\{T\}(t)][T(t)]^\{-1\}\backslash textbf\{P\}(t)\; =\; \backslash begin\{Bmatrix\}\; \backslash textbf\{V\}\_P\; \backslash \backslash \; 0\backslash end\{Bmatrix\}\; =\; \backslash begin\{bmatrix\}\; \backslash dot\{A\}A^T\; \&\; -\backslash dot\{A\}A^T\backslash textbf\{d\}\; +\; \backslash dot\{\backslash textbf\{d\}\}\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\{bmatrix\}$
\begin{Bmatrix} \textbf{P}(t) \\ 1\end{Bmatrix}=[S]\textbf{P}.
The matrix [S] is given by
- $[S]\; =\; \backslash begin\{bmatrix\}\; \backslash Omega\; \&\; -\backslash Omega\backslash textbf\{d\}\; +\; \backslash dot\{\backslash textbf\{d\}\}\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\{bmatrix\}$
where
- $[\backslash Omega]\; =\; \backslash dot\{A\}A^T,$
is the angular velocity matrix.
Multiplying by the operator [S], the formula for the velocity V_{P} takes the form
- $\backslash textbf\{V\}\_P\; =\; [\backslash Omega](\backslash textbf\{P\}-\backslash textbf\{d\})\; +\; \backslash dot\{\backslash textbf\{d\}\}\; =\; \backslash omega\backslash times\; \backslash textbf\{R\}\_\{P/O\}\; +\; \backslash textbf\{V\}\_O,$
where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω], the vector
- $\backslash textbf\{R\}\_\{P/O\}=\backslash textbf\{P\}-\backslash textbf\{d\},$
is the position of P relative to the origin O of the moving frame M, and
- $\backslash textbf\{V\}\_O=\backslash dot\{\backslash textbf\{d\}\},$
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,
- $\backslash textbf\{A\}\_P\; =\; \backslash frac\{d\}\{dt\}\backslash textbf\{V\}\_P\; =\; \backslash frac\{d\}\{dt\}\backslash big([S]\backslash textbf\{P\}\backslash big)=[\backslash dot\{S\}]\backslash textbf\{P\}\; +\; [S]\backslash dot\{\backslash textbf\{P\}\}\; =\; [\backslash dot\{S\}]\backslash textbf\{P\}\; +\; [S][S]\backslash textbf\{P\}\; .$
This equation can be expanded by first computing
- $[\backslash dot\{S\}]\; =\; \backslash begin\{bmatrix\}\; \backslash dot\{\backslash Omega\}\; \&\; -\backslash dot\{\backslash Omega\}\backslash textbf\{d\}\; -\backslash Omega\backslash dot\{\backslash textbf\{d\}\}\; +\; \backslash ddot\{\backslash textbf\{d\}\}\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; \backslash dot\{\backslash Omega\}\; \&\; -\backslash dot\{\backslash Omega\}\backslash textbf\{d\}\; -\backslash Omega\backslash textbf\{V\}\_O\; +\; \backslash textbf\{A\}\_O\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\{bmatrix\}$
and
- $[S]^2\; =\; \backslash begin\{bmatrix\}\; \backslash Omega\; \&\; -\backslash Omega\backslash textbf\{d\}\; +\; \backslash textbf\{V\}\_O\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\{bmatrix\}^2\; =\; \backslash begin\{bmatrix\}\; \backslash Omega^2\; \&\; -\backslash Omega^2\backslash textbf\{d\}\; +\; \backslash Omega\backslash textbf\{V\}\_O\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\{bmatrix\}.$
The formula for the acceleration A_{P} can now be obtained as
- $\backslash textbf\{A\}\_P\; =\; \backslash dot\{\backslash Omega\}(\backslash textbf\{P\}\; -\; \backslash textbf\{d\})\; +\; \backslash textbf\{A\}\_O\; +\; \backslash Omega^2(\backslash textbf\{P\}-\backslash textbf\{d\}),$
or
- $\backslash textbf\{A\}\_P\; =\; \backslash alpha\backslash times\backslash textbf\{R\}\_\{P/O\}\; +\; \backslash omega\backslash times\backslash omega\backslash times\backslash textbf\{R\}\_\{P/O\}\; +\; \backslash textbf\{A\}\_O,$
where α is the angular acceleration vector obtained from the derivative of the angular velocity matrix,
- $\backslash textbf\{R\}\_\{P/O\}=\backslash textbf\{P\}-\backslash textbf\{d\},$
is the relative position vector, and
- $\backslash textbf\{A\}\_O\; =\; \backslash ddot\{\backslash textbf\{d\}\}$
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. Constraints can also arise from other interactions such as rolling without slipping, is any condition relating properties of a dynamic system that must hold true at all times.
Below 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,
- $\backslash boldsymbol\{\; v\}\_G(t)\; =\; \backslash boldsymbol\{\backslash Omega\}\; \backslash times\; \backslash boldsymbol\{\; r\}\_\{G/O\}.$
For the case of an object that does not tip or turn, this reduces to $v\; =\; r\; \backslash omega$.
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.^{[14]}^{[15]}^{[16]} 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.^{[17]} An equilibrium problem (not kinematic) of this type is the catenary.^{[18]}
Kinematic pairs
Main article:
kinematic pair
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^{[19]} 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. We have 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 plan 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, 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, or links, connected by kinematic pairs, or joints, are called 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 of kinematic chains: The planar one degree-of-freedom linkages assembled from N links and j hinged or sliding joints are:
- N=2, j=1: this is a two-bar linkage known as the lever;
- N=4, j=4: this is the four-bar linkage;
- N=6, j=7: this is a six-bar linkage. A six-bar linkage must have two links that support three joints, called ternary links. 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;^{[20]}
- N=8, j=10: the eight-bar linkage has 16 different topologies;
- N=10, j=13: the 10-bar linkage has 230 different topologies,
- N=12, j=16: the 12-bar has 6856 topologies.
See Sunkari and Schmidt^{[21]} for the number of 14- and 16-bar topologies, as well as the number of linkage topologies that have two, three and four degrees-of-freedom.
See also
Notes
References
Further reading
External links
- Java applet of 1D kinematics
- Physclips: Mechanics with animations and video clips from the University of New South Wales
- e-book library of classic texts on mechanical design and engineering.
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