Linear and Angular Velocities 2/4. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

Transcription

1 Linear and ngular elocities 2/4 Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

2 Jacobian Matri - alculation Methods Differentiation the Forward Kinematics Eqs. Iterative ropagation (elocities or Forces / Torques) Jacobian Matri Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

3 Jacobian Matri - Introduction In the field of robotics the Jacobian matri describe the relationship between the joint angle rates ( ) and the translation and rotation velocities of the end effector ( ). This relationship is given b: N J In addition to the velocit relationship, we are also interested in developing a relationship between the robot joint torques ( forces and moments ( ) and the ) at the robot end effector (Static onditions). This relationship is given b: F F T J F Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

4 elocit ropagation Link / Joint bstraction Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

5 elocit ropagation Intuitive Eplanation Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

6 elocit ropagation Intuitive Eplanation Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

7 entral Topic - Simultaneous Linear and otational elocit ector Form Matri Form OG OG ),,,, ( f OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

8 Definitions - Linear elocit Linear velocit - The instantaneous rate of change in linear position of a point relative to some frame. t t t t t t t t dt d t t ) ( ) ( lim ) ( ) ( lim OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

9 Definitions - Linear elocit The position of point in frame {} is represented b the linear position vector The velocit of a point relative to frame {} is represented b the linear velocit vector dt d OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

10 OG OG Linear elocit - igid od Given: onsider a frame {} attached to a rigid bod whereas frame {} is fied. The orientation of frame {} with respect to frame {} is not changing as a function of time roblem: describe the motion of of the vector relative to frame {} Solution: Frame {} is located relative to frame {} b a position vector and the rotation matri OG (assume that the orientation is not changing in time ) epressing both components of the velocit in terms of frame {} gives OG ( ) OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

11 Linear elocit Translation Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

12 Linear elocit Translation Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

13 Linear & ngular elocities - Frames When describing the velocit (linear or angular) of an object, there are two important frames that are being used: epresented Frame (eference Frame) : This is the frame used to represent (epress) the object s velocit. omputed Frame This is the frame in which the velocit is measured (differentiate the position). OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

14 Frame - elocit s with an vector, a velocit vector ma be described in terms of an frame, and this frame of reference is noted with a leading superscript. velocit vector computed in frame {} and represented in frame {} would be written dt d ) ( omputed (Measured) epresented (eference Frame) OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

15 Frame - Linear elocit We can alwas remove the outer, leading superscript b eplicitl including the rotation matri which accomplishes the change in the reference frame Note that in the general case because ma be time-verging If the calculated velocit is written in terms of of the frame of differentiation the result could be indicated b a single leading superscript. In a similar fashion when the angular velocit is epresses and measured as a vector ) ( ) ( ) ( ) ( OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

16 OG OG Frames - Linear elocit - Eample Given: The driver of the car maintains a speed of 1 km/h (as shown to the driver b the car s speedometer). roblem: Epress the velocities in each section of the road,,, D, E, F where {} - ar frame, and {W} - World frame W W W W epresented (eference Frame) {} D omputed (Measured) {W} E Object Frame F Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

17 OG OG Frames - Linear elocit - Eample c s ot( ˆ, ) s c o ot( ˆ, 45 ) o ot( ˆ, 45 ) o ot( ˆ, 9 ) Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

18 Frames - Linear elocit - Eample is not time-varing (in this eample) ) ( [] [] ) ( I W W W ) ( W W W W W W I ) ( [] [] ) ( W W W OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

19 Frames - Linear elocit - Eample oad Section elocit W W W W D E F OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

20 OG OG Linear elocit - Free ector Linear velocit vectors are insensitive to shifts in origin. onsider the following eample: {} {} {} The velocit of the object in {} relative to both {} and {} is the same, that is s long as {} and {} remain fied relative to each other (translational but not rotational), then the velocit vector remains unchanged (that is, a free vector). Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

21 OG OG ngular elocit - igid od Given: onsider a frame {} attached to a rigid bod whereas frame {} is fied. The vector is constant as view from frame {} roblem: describe the velocit of the vector representing the the point relative to frame {} Solution: Even though the vector is constant as view from frame {} it is clear that point will have a velocit as seen from frame {} due to the rotational velocit OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

22 ngular elocit - igid od - Intuitive pproach OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

23 OG OG ngular elocit - igid od - Intuitive pproach The figure shows to instants of time as the vector rotates around This is what an observer in frame {} would observe. The Magnitude of the differential change is sin t Using a vector cross product we get sin sin (t) ( t t) t Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

24 ngular elocit - igid od - Intuitive pproach OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

25 ngular elocit - igid od - Intuitive pproach OG OG otation in 2D Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

26 ngular elocit - igid od - Intuitive pproach In the general case, the vector ma also be changing with respect to the frame {}. dding this component we get. Using the rotation matri to remove the dual-superscript, and since the description of at an instance is we get OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

27 OG OG Definitions - ngular elocit ngular elocit: The instantaneous rate of change in the orientation of one frame relative to another. Linear elocit oint - 1D lane - 2D / od - 3D ngular elocit lane - 2D / od - 3D Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

28 OG OG Definitions - ngular elocit Just as there are man was to represent orientation (Euler ngles, oll-itch- Yaw ngles, otation Matrices, etc.) there are also man was to represent the rate of change in orientation. ngular elocit epresentation ngular elocit ector ngular elocit Matri The angular velocit vector is convenient to use because it has an eas to grasp phsical meaning. However, the matri form is useful when performing algebraic manipulations. Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

29 Definitions - ngular elocit - ector ngular elocit ector: vector whose direction is the instantaneous ais of rotation of one frame relative to another and whose magnitude is the rate of rotation about that ais. The angular velocit vector describes the instantaneous change of rotation of frame {} relative to frame {} OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

30 Definitions - ngular elocit - Matri The rotation matri ( ) defines the orientation of frame {} relative to frame {}. Specificall, the columns of are the unit vectors of {} represented in {}. If we look at the derivative of the rotation matri, the columns will be the velocit of each unit vector of {} relative to {}: dt d OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

31 Definitions - ngular elocit - Matri The relationship between the rotation matri and the derivative of the rotation matri can be epressed as follows: where is defined as the angular velocit matri OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

32 ngular elocit - Matri & ector Forms Matri Form ector Form Definition Multipl b onstant Multipl b ector Multipl b Matri k k T s t s t s t OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

33 Simultaneous Linear and otational elocit - ector ersus Matri epresentation ector Form Matri Form OG OG ) ( ) ( ) ( k j i k j i OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

34 Simultaneous Linear and otational elocit The final results for the derivative of a vector in a moving frame (linear and rotation velocities) as seen from a stationar frame ector Form Matri Form OG OG Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

35 hanging Frame of epresentation - Linear elocit We have alread used the homogeneous transform matri to compute the location of position vectors in other frames: T To compute the relationship between velocit vectors in different frames, we will take the derivative: d dt d T dt T T Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

36 hanging Frame of epresentation - Linear elocit ecall that so that the derivative is 1 org T 1 org org org dt d T T T Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

37 hanging Frame of epresentation - Linear elocit Substitute the previous results into the original equation we get This epression is equivalent to the following three-part epression: org T T T 1 1 org org T T org Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

38 hanging Frame of epresentation - Linear elocit onverting from matri to vector form ields org org Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

39 hanging Frame of epresentation - ngular elocit We use rotation matrices to represent angular position so that we can compute the angular position of {} in {} if we know the angular position of {} in {} and {} in {} b To derive the relationship describing how angular velocit propagates between frames, we will take the derivative Substituting the angular velocit matries we find Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

40 hanging Frame of epresentation - ngular elocit ost-multipling both sides b,which for rotation matrices, is equivalent to The above equation provides the relationship for changing the frame of representation of angular velocit matrices. The vector form is given b To summarie, the angular velocities of frames ma be added as long as the are epressed in the same frame. T T 1 T T T Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL

41 Summar hanging Frame of epresentation Linear and otational elocit ector Form Matri Form ngular elocit ector Form Matri Form OG OG T Instructor: Jacob osen dvanced obotic - ME 263D - Department of Mechanical & erospace Engineering - UL