Degrees of freedom (mechanics) in the context of Motor program

An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downward, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger.

A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position.

Six degrees of freedom (6DOF), or sometimes six degrees of movement, refers to the six mechanical degrees of freedom of movement of a rigid body in three-dimensional space. Specifically, the body is free to change position as forward/backward (surge), up/down (heave), left/right (sway) translation in three perpendicular axes, combined with changes in orientation through rotation about three perpendicular axes, often termed yaw (normal axis), pitch (transverse axis), and roll (longitudinal axis).

Three degrees of freedom (3DOF), a term often used in the context of virtual reality, typically refers to tracking of rotational motion only: pitch, yaw, and roll.

A flexure bearing is a category of flexure which is engineered to be compliant in one or more angular degrees of freedom. Flexure bearings are often part of compliant mechanisms. Flexure bearings serve much of the same function as conventional bearings or hinges in applications which require angular compliance. However, flexures require no lubrication and exhibit very low or no friction.

Many flexure bearings are made of a single part: two rigid structures joined by a thin "hinge" area. A hinged door can be created by implementing a flexible element between a door and the door frame, such that the flexible element bends allowing the door to pivot open.

A revolute joint (also called pin joint or hinge joint) is a one-degree-of-freedom kinematic pair used frequently in mechanisms and machines. The joint constrains the motion of two bodies to pure rotation along a common axis. The joint does not allow translation, or sliding linear motion, a constraint not shown in the diagram. Almost all assemblies of multiple moving bodies include revolute joints in their designs. Revolute joints are used in numerous applications such as door hinges and other uni-axial rotation devices.

A revolute joint is usually made by a pin or knuckle joint, through a rotary bearing. It enforces a cylindrical contact area, which makes it a lower kinematic pair, also called a full joint. However, If there is any clearance between the pin and hole (as there must be for motion), so-called surface contact in the pin joint actually becomes line contact.

In geometry, there exist various rotation formalisms to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

According to Euler's rotation theorem, the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom.

In animal anatomy, a pivot joint (trochoid joint, rotary joint or lateral ginglymus) is a type of synovial joint whose movement axis is parallel to the long axis of the proximal bone, which typically has a convex articular surface.

According to one classification system, a pivot joint like the other synovial joint—the hinge joint has one degree of freedom. Note that the degrees of freedom of a joint is not the same as a joint's range of motion.

A hinge joint (ginglymus or ginglymoid) is a bone joint where the articular surfaces are molded to each other in such a manner as to permit motion only in one plane. According to one classification system they are said to be uniaxial (having one degree of freedom).

The direction which the distal bone takes in this motion is rarely in the same plane as that of the axis of the proximal bone; there is usually a certain amount of deviation from the straight line during flexion.

Ludwig Ernst Hans Burmester (5 May 1840 – 20 April 1927) was a German kinematician and geometer.

His doctoral thesis Über die Elemente einer Theorie der Isophoten (from German: About the elements of a theory of isophotes) concerned lines on a surface defined by light direction. After a period as a teacher in Łódź he became professor of synthetic geometry at Dresden where his growing interest in kinematics culminated in his Lehrbuch der Kinematik, Erster Band, Die ebene Bewegung (Textbook of Kinematics, First Volume, Planar Motion) of 1888, developing the approach to the theory of linkages introduced by Franz Reuleaux, whereby a planar mechanism was understood as a collection of Euclidean planes in relative motion with one degree of freedom. Burmester considered both the theory of planar kinematics and practically all actual mechanisms known in his time. In doing so, Burmester developed Burmester theory which applies projective geometry to the loci of points on planes moving in straight lines and in circles, where any motion may be understood in relation to four Burmester points.

Kinematic coupling describes fixtures designed to exactly constrain the part in question, providing precision and certainty of location. A canonical example of a kinematic coupling consists of three radial v-grooves in one part that mate with three hemispheres in another part. Each hemisphere has two contact points for a total of six contact points, enough to constrain all six of the part's degrees of freedom. An alternative design consists of three hemispheres on one part that fit respectively into a tetrahedral dent, a v-groove, and a flat.

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into${\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}\ ,}$where ${\displaystyle \left\{\ (P_{1},Q_{1}),\ (P_{2},Q_{2}),\ (P_{3},Q_{3}),\ \ldots \ \right\}}$ are the new co‑ordinates, grouped in canonical conjugate pairs of momenta ${\displaystyle P_{i}}$ and corresponding positions ${\displaystyle Q_{i},}$ for ${\displaystyle i=1,2,\ldots \ N,}$ with ${\displaystyle N}$ being the number of degrees of freedom in both co‑ordinate systems.