18 Questions
What is the number of degrees of freedom (DOF) for a rigid body in planar mechanisms?
3
What type of mechanism has a closed loop, like a person standing with both feet on the ground?
Closed-chain mechanism
What is the formula used to calculate the degree of freedom of a mechanism?
Grübler’s formula
What is the number of degrees of freedom (DOF) for a rigid body in spatial mechanisms?
6
What is the type of joint that provides one degree of freedom?
Revolute joint
What is the condition for Grübler’s formula to hold?
All joint constraints are independent
What is the purpose of loop-closure equations in robotics?
To reduce the dimension of the configuration space
What is the configuration space of a robot with n joints and k independent constraints?
A surface of dimension n-k in R^n
What is the relationship between the number of joints and the number of constraints in a robot?
The number of joints is always greater than the number of constraints
What is the purpose of differentiating both sides of the loop-closure equations with respect to time?
To obtain the velocity constraints of the robot
What type of constraint reduces the dimension of the configuration space?
Holonomic constraint
What is the representation of the configuration space of a robot?
A column vector θ = [θ1 · · · θn]!
What is the total number of joints in the five-bar planar linkage shown in Figure 2.5 (b)?
4
What is the degree of freedom of the mechanism shown in Figure 2.5 (c)?
1
What is the total number of joints in the Stewart–Gough platform?
18
What is the degree of freedom of the Stewart–Gough platform?
6
What is the effect of replacing universal joints with spherical joints in the Stewart–Gough platform?
Increases the degree of freedom by 1
What is the application of the Stewart–Gough platform?
Car and airplane cockpit simulators
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.
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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