Mechanisms: Degree Of Freedom

Mechanisms Degree of freedom

Planar and Spatial Mechanisms • Mechanisms can be divided into planar mechanisms and spatial mechanisms, according to the relative motion of the rigid bodies. • In a planar mechanisms, all of the relative motions of the rigid bodies are in one plane or in parallel planes. • If there is any relative motion that is not in the same plane or in parallel planes, the mechanism is called the spatial mechanism

Planar mechanisms • A majority of mechanisms exhibit motion such that the parts move in parallel planes. • Below two identical mechanisms are used on opposite sides of the platform for stability. • The motion of these mechanisms is strictly in the vertical plane. • Therefore, these mechanisms are called planar mechanisms because their motion is limited to two-dimensional space. • Most commercially produced mechanisms are planar .

Planer Mechanism • There are two kinds of pairs in Planer mechanisms , lower pairs and higher pairs. • What differentiates them is the type of contact between the two bodies of the pair. • Surface-contact pairs are called lower pairs(full joint). • In planar (2D) mechanisms, there are two subcategories of lower pairs -- revolute pairs and prismatic pairs.

Planer Mechanism • Point-, line-, or curve-contact pairs are called higher pairs (half joint).

Degree of freedom • Def1: DOF is the minimum number of independent variables required to describe a system configuration completely. • Def2 : no. of independent motions.

DOF • The mobility of a mechanism is its number of degrees of freedom. • DOF is the most important thing of a mechanism • It can also be defined as the number of actuators needed to operate the mechanism. • A mechanism actuator could be manually moving one link to another position, connecting a motor to the shaft of one link, or pushing a piston of a hydraulic cylinder. • Most commercially produced mechanisms have one degree of freedom. In contrast , robotic arms can have three, or more, degrees of freedom.

Configuration variables • DOF is the minimum number of independent variables required to describe a system configuration completely. • The set of variables (dependent or independent) used to describe a system configuration are termed as the configuration variables. • For a mechanism, these can be either Cartesian coordinates of certain points on the mechanism, or the joint angles of the links, or a combination of both.

Configuration space • The set of configuration variables form what is known as the configuration space (denoted by C) of the mechanism. • The degrees-of-freedom of a mechanical system (denoted by M) may or may not equal the dimension of C (denoted by dim(c))

DOF of a point in a plane • Definition :The number of independent coordinates required to define its position • Consider a particle free to move in the XY plane. • Clearly, the particle has two degrees-of-freedom, namely: the two independent translations in the plane. • These can be completely described by the Cartesian coordinates (x, y), or the planar polar coordinates (r, φ), where: x = r cos φ ,y = r sin φ • For such a unconstrained system, it is obvious that M = dim(C).

DOF of a rigid body in a plane • Definition: It’s the no. of independent motions. • Consider two links AB and CD in a plane motion. • The link AB is a reference link (or fixed link). • The configuration (position and orientation) of a link CD can be completely specified by the three variables. i.e. the coordinates of P denoted by x and y, and inclination θ of link CD w.r.t. x-axis or link AB.

DOF of a rigid body in a plane • Any independent rigid body in a plane has 3 DOF in planar motion i.e two translational and one rotational. • Therefore, a system of L unconnected links in the same plane will have 3L DOF, • And the system has a total of six DOF.

DOF of a kinematic pair in a plane • When two links are connected by a revolute joint in as in Figure, ΔY1 and ΔY2 are combined as ΔY, and Δx1 and Δx2 are combined as Δx . This removes two DOF, leaving four DOF. • The two links connected by a revolute joint have four DOF

DOF of a full joint in a plane • Two unconnected links: 6 DOF(each link has 3 DOF) • When connected by a full joint (Lower pair): 4 DOF(each full joint eliminates 2 DOF)

DOF of a full joint in a plane • The revolute and prismatic joints make up all low-pair joints in planar mechanisms. • The observational results can be expressed as a rule: a low-pair joint reduces the mobility of a mechanism by two DOF

DOF of a half joint in a plane • The half joint(higher pair) removes only one DOF from the system (because a half joint has two DOF), leaving the system of two links connected by a half joint with a total of five DOF.

DOF of a half joint in a plane • if disconnected, the system will have six DOF; • if connected by a high-pair joint, it will have five DOF. • This can be stated as a rule: a high-pair joint reduces the mobility of a mechanism by one DOF.

DOF of a mechanism in a plane • Def3: Degree of freedom (also called the mobility M) of a system can be defined as : the number of inputs which need to be provided in order to create a predictable output. • When one link is a reference link ,the kinematic chain is called mechanism. • when a link is connected to a fixed link/another link of mechanism by a turning pair (i.e. lower pair), two degrees of freedom are destroyed.

Example

Example

Determining DOF’s of a Planer Linkage/Mechanism

• Now let us consider a plane mechanism with L number of links. • Since in a mechanism, one of the links is to be fixed, therefore the number of movable links will be (L - 1) • The total number of degrees of freedom will be 3 (L - 1) before they are connected to other link.

Determining DOF’s • In general, a mechanism with L number of links connected by j number of binary joints or lower pairs (i.e. single degree of freedom pairs) and h number of higher pairs (i.e. two degree of freedom pairs), then the number of degrees of freedom of a mechanism is given by M = 3 (L - 1) - 2j – h • This equation is called Gruebler’s criterion/Kutzbach’s criterion of mobility for the movability of a mechanism having plane motion. • If there are no two degree of freedom pairs (i.e. higher pairs), then h = 0. Substituting h = 0 in equation, we have M = 3 (L - 1) - 2j

Gruebler’s/Kutzbach’s equation • It should be noted that Gruebler’s/Kutzbach’s equation has no information in it about link sizes or shapes, only their quantity.

Mechanisms and Structures

Multi DOF linkages • Linkages with multiple degrees of freedom need more than one driver to precisely operate them. • Common multi-degree-of-freedom mechanisms are open-loop kinematic chains used for reaching and positioning, such as robotic arms.