Configuration Space PDF

1/2/23 CONFIGURATION SPACE Week 4.1 IE454– Industrial Robotics Dr. Majed Moosa 1 NEXT GENERATIONS ROBOTS: 2 2 1 1/2/23 LOCALIZATION The most fundamental question you can ask about a robot is "Where is it?” The answer to this question is: The Robot’s Configuration which is a specification...

1/2/23 CONFIGURATION SPACE Week 4.1 IE454– Industrial Robotics Dr. Majed Moosa 1 NEXT GENERATIONS ROBOTS: 2 2 1 1/2/23 LOCALIZATION The most fundamental question you can ask about a robot is "Where is it?” The answer to this question is: The Robot’s Configuration which is a specification of the positions of all the points of the robot The configuration space, which we often call the C-space for short, is the space of all configurations of the robot 3 4 2 1/2/23 DEGREES OF FREEDOM OF A ROBOT The number of independent joint variables required to specify the location of all the links of the robot in space. It is equal to the minimal number of actuated joints to control the system. 5 TYPICAL ROBOT JOINTS Every joint connects exactly two links Joints that simultaneously connect three or more links are not allowed. 6 3 1/2/23 THE NUMBER OF DEGREES OF FREEDOM F AND CONSTRAINTS C PROVIDED BY COMMON JOINTS. 7 GRUBLER'S FORMULA § The number of degrees of freedom of a mechanism with links and joints can be calculated using Grubler's formula.. § dof= § N=# of bodies, including ground § J=# of joints § m = 3 for planar mechanisms and m = 6 for spatial mechanisms), fi be the Grubler's Formula 8 4 1/2/23 EXAMPLES Four-bar linkage. • The planar four bar linkage consists of four links (one of them ground) arranged in a single closed loop and connected by four revolute joints. Since all the links are conned to move in the same plane, we have m = 3. Subsituting N = 4, J = 4, and fi = 1, i = 1 ; …. ; 4, into Grubler's formula, we see that the four-bar linkage has one degree of freedom. Slider-crank mechanism. The slider-crank closed-chain mechanism consists of three revolute joints and one prismatic joint (J = 4 and each fi = 1) and four links (N = 4, including the ground link), Subsituting into Grubler's formula, we see that the four-bar linkage has one degree of freedom. 9 10 5 1/2/23 EXAMPLE • N= • J= • m= • f= • DOF= 11 SOLUTION The planar mechanism has three links that meet at a single point on the right of the large link. Recalling that a joint by definition connects exactly two links, the joint at this point of intersection should not be regarded as a single revolute joint. Rather, it is correctly interpreted as two revolute joints overlapping each other. Again, there is more than one way to derive the number of degrees of freedom of this mechanism using Grubler's formula: (i) The mechanism consists of eight links (N = 8), eight revolute joints, and one prismatic joint. Substituting into Grubler's formula yields dof = 3(8 - 1 - 9) + 9(1) = 3 12 6 1/2/23 IN CLASS EXERCISE 1 • Find DOF for k-link planar serial chain using Grubler's formula ? 13 SOLUTION 1 • N = 5 (four links plus ground), J = 5, and since all joints are revolute, each fi = 1. Therefore: dof = 3(5 - 1 - 5) + 5 = 2 14 7 1/2/23 IN CLASS EXERCISE 2 • Find DOF for the following using Grubler's formula ? 15 SOLUTION 2 • we have N = 6, J = 7, and fi = 1 for all i , so that: dof = 3(6 - 1 - 7) + 7 = 1 16 8