Chapter 1: Degrees of freedom (mechanics)
The degrees of freedom (DOF) of a mechanical system are the number of independent characteristics that describe the configuration or state of the system. This concept originates from the field of physics. It plays a significant role in the analysis of body systems in a variety of domains, including mechanical engineering, structural engineering, aerospace engineering, robotics, and others.
There is one degree of freedom in the position of a single railcar (engine) that is going along a track. This is due to the fact that the position of the car is determined by the distance along the track. There is still only one degree of freedom available to a train consisting of rigid cars that are attached to an engine by hinges. This is due to the fact that the positions of the cars behind the engine are restricted by the form of the track.
An automobile that has a suspension that is extremely rigid might be thought of as a rigid body that is transporting itself on a plane, which is a two-dimensional region that is flat. This body possesses three degrees of freedom that are independent of one another. These degrees of freedom are comprised of two components of translation and one angle of rotation. The three separate degrees of freedom that an automobile possesses are best shown by the act of skidding or drifting.
There are six degrees of freedom for a rigid body because its position and orientation in space are determined by three components of translation and three components of rotation. This determines that the rigid body has six degrees of freedom.
When it comes to mechanical design, the precise constraint technique is responsible for managing degrees of freedom in such a way that a device is neither under- or overdesigned.
The position of an n-dimensional rigid body is determined by the rigid transformation, denoted as [T] = [A, d]. In this equation, d represents an n-dimensional translation, and A is a n? n rotation matrix. The matrix A possesses n degrees of freedom in the translational direction, and n degrees of freedom in the rotating direction, which is equal to n(n-1)/2. It is the dimension of the rotation group SO(n) that determines the number of degrees of freedom that are associated with rotation.
It is possible to think of a non-rigid or flexible body as a collection of many minute particles (an infinite number of degrees of freedom), and a finite degree of freedom system is frequently used to approximate this concept. It is possible to simplify the analysis by approximating a deformable body as a rigid body (or even a particle) in situations when the primary purpose of the study is to investigate motion that involves huge displacements. For example, when evaluating the motion of satellites, this would be the case.
One way to think about the degree of freedom of a system is as the bare minimum of coordinates that are necessary to express a configuration. Therefore, if we apply this definition, we have:
Three translations (3T) and three rotations (3R) make up the three degrees of freedom (3T3R) that a single rigid body can have, with a maximum of six degrees of freedom (6 DOF).
Check out Euler angles as well.
A ship's motion at sea, for instance, is characterized by the six degrees of freedom that are associated with a rigid body and can be stated as follows:
For instance, the trajectory of an airplane while it is in flight has three degrees of freedom, and the attitude of the airplane along the trajectory also has three degrees of freedom, making the total number of degrees of freedom from three to six.
It's possible that the amount of degrees of freedom that a single rigid body can have is restricted by physical limits. To give an example, a block that is sliding about on a flat surface has three degrees of freedom two translations and one rotation, which is denoted by the notation 2T1R. SCARA is an example of an XYZ positioning robot that has three degrees of freedom and three degrees of motion.
The mobility formula is a formula that counts the number of factors that characterize the configuration of a collection of rigid bodies that are confined by joints linking these bodies.
Take into consideration a system consisting of n rigid bodies that are at rest in space and have 6n degrees of freedom when measured in relation to a fixed frame. It is necessary to include the fixed body in the count of bodies in order to determine the number of degrees of freedom that are associated with this system. This will ensure that mobility is not contingent on the selection of the body that constitutes the fixed frame. Consequently, the degree of freedom of the unconstrained system with the equation N = n + 1 is
mainly due to the fact that the immovable body possesses zero degrees of freedom in relation to them.
Within this system, the joints that connect the bodies eliminate degrees of freedom and hence decrease mobility. To be more specific, hinges and sliders each impose five limits, and as a result, they eliminate five degrees of freedom. A handy method to define the number of constraints c that a joint imposes is to do so in terms of the joint's freedom f, where c is equal to six times f minus 6 times f. Given that a hinge or slider, both of which are joints with one degree of freedom, have f equal to one, it follows that c equals six times one, which is five.
The conclusion that can be drawn from this is that the mobility of a system that is composed of n moving links and j joints, each of which has freedom fi, where i = 1,..., j, is specified by
You should keep in mind that N contains the fixed link.
Both a simple open chain and a simple closed chain are considered to be essential particular examples. First, there is a simple open chain.
There are n moving links that are joined to one another by n joints, and one of the ends of the chain is connected to a ground link. This is the single open chain. For this reason, the value of N is equal to j plus one, and the mobility of the chain equals
n moving links are joined end-to-end by n plus one joints to form a simple closed chain. This is done in such a way that the two ends of the chain are connected to the ground link, so becoming a loop. This particular instance is one in which N equals j, and the mobility of the chain is
For instance, a serial robot manipulator is an example of a straightforward open chain. By virtue of the fact that these robotic systems are built from a series of links that are joined by six revolute or prismatic joints that each have one degree of freedom, the system possesses a total of six degrees of freedom.
For instance, the RSSR spatial four-bar linkage is an illustration of a straightforward closed chain. Due to the fact that the total number of degrees of freedom for these joints is eight, the mobility of the linkage is two. One of the degrees of freedom is the rotation of the coupler around the line that connects the two S joints.
In order to create what is known as a planar linkage, it is usual practice to build the linkage system in such a way that the movement of all of the bodies is limited to lay on parallel planes that are parallel to one another. A spherical linkage can also be created by constructing the linkage system in such a way that all of the bodies move on concentric spheres. This too has the potential to be completed. This means that the degrees of freedom of the links in each system are now three instead of six, and the constraints that are imposed by joints are now in the form of c = 3 minus f. This is the case in both cases.
The formula for mobility is given by in this particular instance.
as well as the exceptional circumstances
The planar four-bar linkage is an example of a planar simple closed chain. This linkage is a four-bar loop that has four joints that each have one degree of freedom, and as a result, it has mobility M equal to 1.
A combined degree of freedom (DOF) would be the sum of the degrees of freedom (DOFs) of the bodies in a system that contains several bodies, minus any internal limitations that the bodies might have on relative motion. The number of degrees of freedom that a single rigid body possesses may be exceeded by a mechanism or linkage that is composed of multiple rigid bodies that are coupled to one another. When referring to the number of factors that are required to determine the spatial pose of a connection, the phrase "degrees of freedom" is also utilized in this context. In addition to this, it is defined in relation to the configuration space, task space, and workspace of a robot.
In the open kinematic chain, a set of stiff links are connected at joints; a joint may allow one degree of freedom (hinge/sliding) or two degrees of freedom (cylindrical). This particular sort of linkage is a special type of linkage. Chains of this kind are frequently seen in the fields of robotics and biomechanics, as well as in satellites and other space structures. 7 degrees of freedom are considered to be present in a human arm. Pitch, yaw, and roll are all functions that can be performed by the shoulder, the elbow, and the wrist. The shoulder is responsible for pitch, yaw, and roll. There would be only three of those motions required to move the hand to any point in space; nevertheless, humans would be unable to grab things from varied angles or directions since they would not be able to accomplish this. Holonomic refers to a robot (or...
