Robotics and Control: Degrees of Freedom (DOF) and Grubler's Formula
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Questions and Answers

What is the number of degrees of freedom (DOF) for a rigid body in planar mechanisms?

  • 6
  • 4
  • 5
  • 3 (correct)

    • What type of mechanism has a closed loop, like a person standing with both feet on the ground?

  • Closed-chain mechanism (correct)
  • Open-chain mechanism
  • Serial mechanism
  • Parallel mechanism

    • What is the formula used to calculate the degree of freedom of a mechanism?

  • Euler’s formula
  • Grübler’s formula (correct)
  • Newton’s formula
  • Kinematic formula

    • What is the number of degrees of freedom (DOF) for a rigid body in spatial mechanisms?

    <p>6</p> Signup and view all the answers

    What is the type of joint that provides one degree of freedom?

    <p>Revolute joint</p> Signup and view all the answers

    What is the condition for Grübler’s formula to hold?

    <p>All joint constraints are independent</p> Signup and view all the answers

    What is the purpose of loop-closure equations in robotics?

    <p>To reduce the dimension of the configuration space</p> Signup and view all the answers

    What is the configuration space of a robot with n joints and k independent constraints?

    <p>A surface of dimension n-k in R^n</p> Signup and view all the answers

    What is the relationship between the number of joints and the number of constraints in a robot?

    <p>The number of joints is always greater than the number of constraints</p> Signup and view all the answers

    What is the purpose of differentiating both sides of the loop-closure equations with respect to time?

    <p>To obtain the velocity constraints of the robot</p> Signup and view all the answers

    What type of constraint reduces the dimension of the configuration space?

    <p>Holonomic constraint</p> Signup and view all the answers

    What is the representation of the configuration space of a robot?

    <p>A column vector θ = [θ1 · · · θn]!</p> Signup and view all the answers

    What is the total number of joints in the five-bar planar linkage shown in Figure 2.5 (b)?

    <p>4</p> Signup and view all the answers

    What is the degree of freedom of the mechanism shown in Figure 2.5 (c)?

    <p>1</p> Signup and view all the answers

    What is the total number of joints in the Stewart–Gough platform?

    <p>18</p> Signup and view all the answers

    What is the degree of freedom of the Stewart–Gough platform?

    <p>6</p> Signup and view all the answers

    What is the effect of replacing universal joints with spherical joints in the Stewart–Gough platform?

    <p>Increases the degree of freedom by 1</p> Signup and view all the answers

    What is the application of the Stewart–Gough platform?

    <p>Car and airplane cockpit simulators</p> Signup and view all the answers

    Study Notes

    Configuration and Velocity Constraints

    • Robots with one or more closed loops can be represented implicitly using loop-closure equations.
    • The four-bar linkage is an example of a closed-loop robot, which can be expressed by three equations.

    Loop-Closure Equations

    • Loop-closure equations are used to represent the configuration space of robots with closed loops.
    • For general robots, the configuration space can be represented by a set of k independent equations, with k ≤ n, where n is the number of unknowns.
    • These constraints are known as holonomic constraints, which reduce the dimension of the C-space (configuration space).

    Closed-Chain Robots

    • A closed-chain robot is a mechanism that has a closed loop, such as a person standing with both feet on the ground.
    • The configuration space of a closed-chain robot can be viewed as a surface of dimension n-k embedded in Rn.

    Grubler's Formula

    • Grubler's formula is used to calculate the degrees of freedom (DOF) of a mechanism.
    • The formula is: DOF = m(N-1-J) + ∑fi, where m is the number of DOF of a rigid body, N is the number of links, J is the number of joints, and fi is the number of freedoms provided by joint i.
    • Examples of mechanisms with DOF calculated using Grubler's formula include the four-bar linkage, slider-crank mechanism, k-link planar serial chain, five-bar planar linkage, Stephenson six-bar linkage, and Watt six-bar linkage.

    Stewart-Gough Platform

    • The Stewart-Gough platform is a mechanism that consists of two platforms connected by six universal-prismatic-spherical (UPS) legs.
    • The DOF of the Stewart-Gough platform can be calculated using Grubler's formula, which gives a result of 6 DOF.
    • In some versions of the Stewart-Gough platform, the universal joints are replaced by spherical joints, which introduces an extra degree of freedom in each leg.

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    Description

    Quiz on degrees of freedom, joints, and constraints in robotics and control, including Grubler's formula and its application to planar and spatial mechanisms.

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