Mekanisme Robot - 3 SKS (Robot Mechanism)

Transcription

1 Mekanisme Robot - 3 SKS (Robot Mechanism) Latifah Nurahmi, PhD!! latifah.nurahmi@gmail.com!! C.250 First Term /2017 Velocity Rate of change of position and orientation with respect to time Linear velocity : V = V x V y V z Angular velocity : ω = ω x ω y ω z Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 2 1

2 Velocity θ Angular velocity of all points on the rigid body : θ = dθ dt Position of point A is R AO, it velocity: VB B VA A V A = ω R AO Position of point B is R BO, it velocity: C y x V B = ω R BO Position of point is R CO, it velocity: V C = ω R CO VC z O Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 3 Velocity Let us recall : OA = OP + PA VA A θ A A = P A + R 3 3. A B Derive w.r.t. time : da A dt = dpa dt + d(r 3 3. A B ) dt V A = V P + ω A A A V A = V P + ω A (R 3 3. A B ) Frame A z y O x Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 4 w u P v Frame B 2

3 Velocity coordination problems Direct velocity problems: input joint rates are known and the aim is to find the EE velocities Inverse velocity problems: EE velocities are known and the aim is to find the input joint rates Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 5 Joint space q: Vector space that is spanned by by the joint input rates End effector space x: Vector space that is spanned by by the EE velocities Jacobian J: Matrix that transforms the joint input rates to the EE velocities Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 6 3

4 Jacobian may lose its full rank This condition is called singularity At singular condition, serial manipulator may lose 1 or more dof Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 7 x i = f i (q 1, q 2, q 3,, q n ) for i = 1,2,3,, m are the set of m equations Each equation has n independent variables Time derivatives of x i are: x i = f i q 1 q 1 + f i q 2 q 2 + f i q 3 q f i q n q n i = 1,2,3,, m Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 8 4

5 Write in the matrix form: Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 9 x = q = x 1, x 2, x 3,, x T m, m dimensional vector q 1, q 2, q 3,, q T n, n dimensional vector q i = θ i d i revolute joint prismatic joint J = m n matrix Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 10 5

6 x can be expressed in 2 different ways: Conventional Jacobian x = v n ω n v n : linear velocity of the origin of EE ω n : angular velocity of the EE Screw-based Jacobian x = ω n v o v o : linear velocity of reference frame on the EE ω n : angular velocity of the EE Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 11 m = n m > n m < n Iso-constrained Kinematically redundant Actuated redundant Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 12 6

7 Conventional Jacobian Any point can be chosen as reference point, usually the origin of EE : O n Linear velocity: Angular velocity: θ i : the rates of rotation along the i-th joint d i : the rates of translation along the i-th joint : unit vector along i-th joint axis : unit vector from the origin of (i-1)th link frame to the origin of EE Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 13 Conventional Jacobian All vectors are expressed in the fixed frame Writing all the equations in matrix form: Where: For revolute joint For prismatic joint Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 14 7

8 Jacobian of the 2-DOF manipulator A coordinate system is attached to each link according to DH convention Compute : Compute : Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 15 Jacobian of the 2-DOF manipulator Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 16 8

9 Jacobian of the 2-DOF manipulator Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 17 Singularity of 2-DOF manipulator Det J = 0 Mekanisme Robot (Robot Mechanisms) by Latifah Nurahmi 18 9