# Chapter 2: Configuration Area

“The configuration of a robot is a whole specification of dobrovol.org the location of every point of the robotic. The minimum numberof actual-valued coordinates had to constitute the configuration is the wide variety of levels of freedom (dof) of the robot. The -dimensional area containing all feasible configuration of the robot is called the configuration area (C-space). The configuration area of a robotic is represented by a point the C-space.”Degrees of Freedom of a Rigid Body

How to decide the dof of a coin on aircraft?Two parameters to determine point $A$One parameter to determine factor $B$ (why?)Point $C$ can determined from $A$ and $B$, so it calls for no parameter.

How about a coin in 3-dimensional area?

For a rigid body in $n$-dimensional area, its dof is $m$, then we’ve got:

in which $n$ stages are linear and all the rest degrees are angular.Constraints of Joints

Joints will constrain the dof of a rigid body.

What are the constrains of several crucial joints(specifically revolute, prismatic, and spherical) in both planar and spatial space?Grübler’s Formula

This formula is to calculate the the quantity of tiers of freedom of a mechanism with links and joints.

It’s based on a popular rule(which can be useful in some cases!): degrees of freedom = (sum of freedoms of the bodies) – (number of unbiased constraints). Remember, right here the constraints need to be independent.

Given a mechanism consisting of $N$ hyperlinks(inclusive of the gound as a hyperlink), $J$ joints. The wide variety of ranges of freedom of a body is $m$, the wide variety of freedoms and constraints supplied via joint $i$ are $f_i$ and $c_i$, Grübler’s formula can written as:Several Notes for Applyging Grübler’s FormulaThe floor is always considered as a link, but its wide variety of stages of freedom is zero.Careful with overlapping joints: instance 2.five, page 20 of the textbook.Careful with redundant constraints and sigularities: instance 2.6, page 21 of the textbook. Recall that Grübler’s method holds most effective if the restrictions are independent, if they are NOT independent, then the end result of Grübler’s components serves because the decrease sure at the dof(In this situation, the dof may be negative). Configuration space sigularities arising in closed chains are mentioned later.The Delta robot: example 2.7, page 22 of the textbook. Even even though the dof calculated from Grübler’s components is 15, however simplest 3 are seen at the quit-effector. This is due to the fact that one diploma of freedom in a spherical joint is for torsion.Some Quiz Questions(Question three of quiz of bankruptcy 2, component 1) Assume your arm, out of your shoulder to your palm, has 7 ranges of freedom. You are carrying a tray like a waiter, and you have to preserve the tray horizontal to keep away from spilling drinks at the tray. How many levels of freedom does your arm have whilst pleasing the constraint that the tray stays horizontal? Your answer should be an integer.The answer is 5.The requirement that the tray be horizontal locations two constraints on its orientation: the rotation of the tray approximately two axes defining the horizontal aircraft of the tray have to be zero. (In different words, the roll and the pitch of the tray are zero.)(Question four&5 of quiz of bankruptcy 2, element 1) A total of $n$ same SRS arms are greedy a not unusual item as shown underneath. Find the wide variety of levels of freedom of this gadget while the grippers hold the item rigidly (no relative motion among the item and the ultimate links of the SRS hands). Your solution ought to be a mathematical expression consisting of $n$.The answer is $n+6$.Note that all the graspers, together with the object, have to be considered as a unmarried hyperlink.(Question 7 of quiz of bankruptcy 2, component 1) Use the planar version of Grubler’s formulation to decide the variety of ranges of freedom of the mechanism proven beneath. Your answer must be an integer.The solution is three.Removing all the redundant joints, this mechanism is equal with an open-chain, 3R arm.Configuration Space: Topology and RepresentationConfiguration Space Topology

In what case, two configuration spaces are topologically equal?

What’s the distinction among $T^2 = S^1 \instances S^1$ and $S^2$?

What’s the full C-area of a rigid frame in three-dimensional area? (Recall the manner of determining the levels of freedom of a inflexible frame in space by way of selecting 3 factors at the frame)Configuration Space Representation

Explicit Representation:Uses $n$ parameters to symbolize a $n$-dimensional area.But it has sigularities, for example the North/South pole state of affairs.There are two solutions for the sigularities:Use more than one coordinate charts (atlas).Use implicit representation.

Implicit Representation:Uses more parameters than the distance’s dof.Has advantage in representing closed-chain mechanismConfiguration and Velocity Constraints

A planar, closed-chain, four-bar linkage has a dof of one, but representing this dof may be tough with the aid of using express representation. With loop-closure equations, however, implicit illustration is extra useful in this example.

Holonomic Constraints:Constraints that reduce the measurement of C-area.The C-space can be considered as a floor of measurement $n-ok$ embedded in $\mathbbR^n$, where $n$ suggests the variety of parameters that are used to represent the space, and $k$ indicates the range of constraints. (In the instance of the planar, closed-chain, four-bar linkage, $n=4$ and $k=three$)

Pfaffian ConstraintsVelocity constraints inside the form of $A(\theta)\dot\theta = 0$ are named Pfaffian constraints. (Where does this shape come from?)Some of them are the by-product of a fixed of holonomic constraints.

Nonholonomic ConstraintsPfaffian constraints which might be nonintegrable. (Figure 2.11: A coin rolling on a aircraft with out slipping)Reduces the measurement of possible velocities of the device, but do now not lessen the accessible C-space. (The rolling coin can reach any vicinity in its 4-dimensional C-area in spite of the 2 constraints on its pace)Task Space and Work Space

Task Space: The space in which the robot’s undertaking is certainly defined.

Work Space: A specification of the on hand configurations of the end-effector.

Both spaces are wonderful from the robot’s C-area.My QuestionsConfiguration Space Singularity in Five-Bar Linkage

In Figure 2.7(b), as shown beneath, if the 2 joints linked to the floor are locked, then the mechanism has a dof of zero. This is true in maximum cases, and may be proven with the aid of Grübler’s formulation. (freedoms of the locked joints and other joints are zero and one)

However, this mechanism has a singularity, as shown within the proper sub-determine. If the two middle links have a identical duration and overlap each other, then thoselinks can rotate together freely.

So, is Grübler’s formula applicable in this example?

My solution is, for this singularity, similar with the mechanism in parent (a), the 2 hyperlinks of one facet (proper or left), together with the 3 joints, haven’t any effect at the motion of the mechanism. Thus, do not forget that $m=3, N=3, J=2$, and $f_1=1, f_2=0$, $dof = m(N-1-J) + f_1 + f_2 = 1$.The Human Arm Problem

For the human arm problem referred to above, the given answer shows that there areconstraints on the palm to maintain the tray horizontal. However, if we count on the arm is a SRS mechanism, that is the belief of the given solution, the dof must be seven. Even though the tray should be horizontal, the spherical joint can nevertheless move in all 3 levels, with the agency with the relaxation two joints. Is my information correct?