Velocity Propagation Between Robot Links 3/4
Velocity Propagation Between Robot Links 3/4Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Introduction – Velocity PropagationInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Jacobian Matrix - Introduction In the field of robotics the Jacobian matrixdescribe the relationship between the jointangle rates ( N ) and the translation androtation velocities of the end effector ( x ).This relationship is given by:x J In addition to the velocity relationship, we arealso interested in developing a relationshipbetween the robot joint torques ( ) and theforces and moments ( F ) at the robot endeffector (Static Conditions). Thisrelationship is given by: J T FInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLAF
Jacobian Matrix - Calculation MethodsDifferentiation theForward Kinematics Eqs.Iterative Propagation(Velocities or Forces / Torques)Jacobian MatrixInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Summary – Changing Frame of Representation Linear and Rotational Velocity– Vector FormA BVQ AVBORG BAR BVQ A B BAR BPQA– Matrix FormVQ AVBORG BAR BVQ BAR BAR BPQ A Angular Velocity– Vector Form– Matrix FormAAC C A B BAR B CR BAR BARCBR BA RTInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLABPQ
Frame - Velocity As with any vector, a velocity vector may be described in terms of any frame,and this frame of reference is noted with a leading superscript. A velocity vector computed in frame {B} and represented in frame {A} would bewrittenQRepresented(Reference Frame)( VQ ) A BAdBPQdtComputed(Measured)VBORG 0A R 0ABInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Position Propagation The homogeneous transform matrix provides a complete description of thelinear and angular position relationship between adjacent links. These descriptions may be combined together to describe the position of a linkrelative to the robot base frame {0}.T o1T 12T i 1i Toi A similar description of the linear and angular velocities between adjacent linksas well as the base frame would also be useful.Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Position Propagation0TT 01T12T23T34T45T56T6TInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLAT
Motion of the Link of a Robot In considering the motion of a robot link we will always use link frame {0} as thereference frame (Computed AND Represented). However any frame can be usedas the reference (represented) frame including the link’s own frame (i)Where:vi i - is the linear velocity of the origin of link frame (i) with respect toframe {0} (Computed AND Represented)- is the angular velocity of the origin of link frame (i) with respect toframe {0} (Computed AND Represented)Expressing the velocity of a frame {i} (associated with link i ) relative to the robotbase (frame {0}) using our previous notation is defined as follows:0 vi 0Vi 0Vi i 0 00iiInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocities - Frame & Notation The velocities differentiate (computed) relative to the base frame {0} are oftenrepresented relative to other frames {k}. The following notation is used for thisconditions R kkkvi 0Vi 0kR 0Vi 0kR vik 0iik00ik0R iInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity Propagation Given: A manipulator - A chain ofrigid bodies each one capable ofmoving relative to its neighbor Problem: Calculate the linear andangular velocities of the link of arobot Solution (Concept): Due to therobot structure (serial mechanism)we can compute the velocities ofeach link in order starting from thebase.The velocity of link i 1 will be that oflink i , plus whatever new velocitycomponents were added by joint i 1Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 0/5Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 1/5 From the relationship developed previouslyA C A B BAR B Cwe can re-assign link names to calculate the velocity of any link i relative to thebase frame {0} A 0 B i C i 1 0 i 1 0 i 0i Ri i 1By pre-multiplying both sides of the equation byof reference for the base {0} to frame {i 1}i 10R ,we can convert the frameInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 2/5i 10 R0 i 1 i 01R0 i i 01R0i Ri i 1Using the recently defined notation, we have i 1 i 1 i i i1Ri i 1i 1 i 1i 1- Angular velocity of frame {i 1} measured relative to the robot base, andexpressed in frame {i 1} - Recall the car example c wV c vcci otbase,and iexpressed in frame {i 1}i 1 i- Angular velocity of frame {i 1} measured relative to frame {i} andi R i 1expressed in frame {i 1}Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 3/5 i 1 i 1 i i i1Ri i 1i 1 Angular velocity of frame {i} measured relative to the robot base, expressed inframe {i 1} i i i1Ri ii 1Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 4/5 i 1 i 1 i i i1Ri i 1i 1 Angular velocity of frame {i 1} measured (differentiate) in frame {i} andrepresented (expressed) in frame {i 1}Assuming that a joint has only 1 DOF. The joint configuration can be eitherrevolute joint (angular velocity) or prismatic joint (Linear velocity).Based on the frame attachment convention in which we assign the Z axispointing along the i 1 joint axis such that the two are coincide (rotations of a linkis preformed only along its Z- axis) we can rewrite this term as follows: 3 2i 1i 1 0 i 1 i i R i 1 0 i 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 5/5 The result is a recursive equation that shows the angular velocity of one link interms of the angular velocity of the previous link plus the relative motion of thetwo links. 0 i 1 i 1 i i1R i i 0 i 1 Since the term i 1 depends on all previous links through this recursion, theangular velocity is said to propagate from the base to subsequent links.i 1Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 0/6Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 1/6 Simultaneous Linear and Rotational Velocity The derivative of a vector in a moving frame (linear and rotation velocities) asseen from a stationary frame Vector FormAVQ AVBORG BAR BVQ A B BAR BPQA Matrix FormVQ AVBORG BAR BVQ BAR BAR BPQ AInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA BBPQ
Velocity of Adjacent Links - Linear Velocity 2/6 From the relationship developed previously (matrix form)VQ AVBORG BAR BVQ BAR BAR BPQ A we re-assign link frames for adjacent links (i and i 1) with the velocitycomputed relative to the robot base frame {0} A 0 B i C i 1 Vi 1 0i R 0i RiPi 1 0Vi 0i RiVi 10 By pre-multiplying both sides of the equation by i 01Rframe of reference for the left side to frame {i 1},we can convert theInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 3/6i 10 R0Vi 1 i 01R0i R 0i RiPi 1 i 01R0Vi i 01R0i RiVi 1Which simplifies toi 10 Factoring outi 1iR0Vi 1 i 01R0i R 0i RiPi 1 i 01R0Vi i i1RiVi 1R from the left side of the first two termsi 10R0Vi 1 i i1R 0i R0i R 0i RiPi 1 0iR0Vi i i1RiVi 1Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 4/6i 10i 1iR0Vi 1 i i1R 0i R0i R 0i RiPi 1 0iR0Vi i i1RiVi 1R iVi 1 - Linear velocity of frame {i 1} measured relative to frame {i} andexpressed in frame {i 1} Assuming that a joint has only 1 DOF. The joint configuration can be eitherrevolute joint (angular velocity) or prismatic joint (Linear velocity). Based on the frame attachment convention in which we assign the Z axispointing along the i 1 joint axis such that the two are coincide (translation of alink is preformed only along its Z- axis) we can rewrite this term as follows: 0 i 1 i i R Vi 1 0 d i 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 5/6 0 i 1 0i 1i 0 0 ii 0 0 R Vi 1 i R 0 R i R i R Pi 1 0 R Vi 0 d i 1 i0R0i R 0i R 0iR0i R 0i RT 0iR0 i 0iR i i iMultiply by MatrixDefinitioni 10DefinitionR0Vi 1 i 1vi 1i0R0Vi i viDefinitionInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 6/6 The result is a recursive equation that shows the linear velocity of one link interms of the previous link plus the relative motion of the two links. 0 i 1vi 1 i i1R i i i Pi 1 i vi 0 d i 1 Since the term i 1vi 1 depends on all previous links through this recursion, theangular velocity is said to propagate from the base to subsequent links.Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Summary Angular Velocity0 - Prismatic Joint 0 i 1 i 1 i i1R i i 0 i 1 Linear Velocity0 - Revolute Joint 0 i 1vi 1 i i1R i i Pi 1 i vi 0 d i 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example For the manipulator shown in the figure, compute the angular and linear velocityof the “tool” frame relative to the base frame expressed in the “tool” frame (thatis, calculate 4 4 and 4 v ).4Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Frame attachmentInstructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example DH Parametersi1234 i 109000ai 10L1L2di000L30Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA i 1 2 30
Angular and Linear Velocities - 3R Robot - Example From the DH parameter table, we can specify the homogeneous transformmatrix for each adjacent link pair: c i s c i 1 i i 1iT s i s i 1 0 c1 s1 s1 c10 1T 0 0 0 0 c 3 s 3 s 3 c32 3T 0 0 0 0 s i0ai 1 c i c i 1 s i 1 s i 1d i c i s i 1 c i 1c i 1d 001 0 0 c 2 s 2 0 L1 00 0 0 1 0 1 2T 1 0 0 0 s2 c2 0 1 00 1 00 L 2 1 0 0 L3 0 1 0 0 0 0 3 T 41 0 0 0 1 0 0 1 0001 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Compute the angular velocity of the end effector frame relative to the baseframe expressed at the end effector frame. 0 i 1 i 1 i i1R i i 0 i 1 Fori 0 0 c1 s1 0 0 0 0 1 1 01R 0 0 0 s1 c1 0 0 0 0 0 1 0 1 1 1 0Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example ForFor For Notei 10 s 2 0 0 s 2 1 0 c2 2 2 12R1 1 0 s 2 0 c 2 0 0 c 2 1 2 0 1 0 1 2 2 i 2 0 c3 s3 0 s 2 1 0 s 23 1 3 3 23R 2 2 0 s3 c3 0 c2 1 0 c23 1 0 1 2 3 2 3 3 0i 3 0 1 0 0 s 23 1 0 s 23 1 4 4 34R 3 3 0 0 1 0 c23 1 0 c23 1 0 0 0 1 2 3 0 2 3 3 4 43Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Compute the linear velocity of the end effector frame relative to the base frameexpressed at the end effector frame.Note that the term involving the prismatic joint has been dropped from theequation (it is equal to zero).0 0 i 1vi 1 i i1R i i Pi 1 i vi 0 d i 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Fori 0 c1 s1 0 0 0 0 0 1v1 01R 0 0 0 P1 0v0 s1 c1 0 0 0 0 0 0 1 0 0 0 0 0 Fori 10 s 2 0 L1 0 0 c2 2v2 12R 1 1 1P2 1v1 s 2 0 c 2 0 0 0 0 0 1 0 1 0 0 L1 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Fori 3s3 0 s 2 1 L 2 0 c3 3v3 23R 2 2 2 P3 2v2 s3 c30 0 c 2 1 0 0 00 1 2 0 L1 1 0L 2s3 2 c3 s 3 0 s3 c3 0 L 2 1L2c3 2 00 1 L 2c 2 1 L1 1 ( L1 L 2c 2) 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Fori 4 L 2s3 2 1 0 0 s 23 1 L3 4v4 34R 3 3 3 P4 3v3 0 1 0 c 23 1 0 L 2c3 2 0 0 1 2 3 0 ( L1 L 2c 2) 1 L 2s3 2 ( L 2c3 L3) 2 L3 3 ( L1 L 2c 2 L3c 23) 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Note that the linear and angular velocities ( 4 4 ,4 v4 ) of the end effector wheredifferentiate (measured) in frame {0} however represented (expressed) in frame{4} In the car example: Observer sitting in the “Car”Observer sitting in the “World” V V C WCW WCkk R kvi 0Vi 0kR 0Vi 0kR vik 0iSolve forik00ik0R iv4 and 4 by multiply both side of the questions from the left by 04R 14v4 04R v4 4 04R 44Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example Multiply both sides of the equation by the inverse transformation matrix, wefinally get the linear and angular velocities expressed and measured in thestationary frame {0}v4 04R 1 4 v4 04RT 4 v4 40R 4 v4 4 04R 1 4 4 04RT 4 4 40R 4 4T 01T 12T 23T 34T04Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example 0 1 1 0 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example s 2 1 2 2 c 2 1 2 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example s 23 1 3 3 c 23 1 2 3 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example 3 4 43Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example 0 1v1 0 0 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example 0 2v2 0 L1 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example L 2s3 2 3v3 L 2c3 2 ( L1 L 2c 2) 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example L 2s3 2 4v4 ( L 2c3 L3) 2 L3 3 ( L1 L 2c 2 L3c 23) 1 Instructor: Jacob RosenAdvanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 4/5 Angular velocity of frame {i 1} measured (differentiate) in frame {i} and represented (expressed) in frame {i 1} Assuming that a joint has only 1 DOF. The joint configuration can be either revolute joint (angular velocity) or prismatic joint (Linear velocity).
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Using the velocity propagation method we expressed the relationship between the velocity of the robot end effector measured relative to the robot base frame {0} and expressed in the end effector frame {N}. Occasionally, it may be desirable to express (represent) the end effector velocities in another frame (e.g. frame
In order to explore the effect of robot types and task types on people s perception of a robot, we executed a 3 (robot types: autonomous robot vs. telepresence robot vs. human) x 2 (task types: objective task vs. subjective task) mixed-participants experiment. Human condition in the robot types was the control variable in this experiment.
1. The robot waits five seconds before starting the program. 2. The robot barks like a dog. 3. The robot moves forward for 3 seconds at 80% power. 4. The robot stops and waits for you to press the touch sensor. 5. The robot moves backwards four tire rotations. 6. The robot moves forward and uses the touch sensor to hit an obstacle (youth can .
steered robot without explicit model of the robot dynamics. In this paper first, a detailed nonlinear dynamics model of the omni-directional robot is presented, in which both the motor dynamics and robot nonlinear motion dynamics are considered. Instead of combining the robot kinematics and dynamics together as in [6-8,14], the robot model is
Fig. 4. Dynamic Analysis of Two Link Robot Arm. The following Fig. 5 is the front view of the experimental setup of the two link robot arm is drawing using the Solid-Wrok. The dimensions of the two link robot arm is expressed in following Table 1; TABLE 1 DIMENSION OF THE 2-LINKS ROBOT ARM Parts of the Robot Arm Dimension of the Parts
PhET Moving Man 1. Set velocity to 0.5 0m/s 3. Set velocity to 2.0 0m/s 2. Set velocity to 1.00m /s 4. Set velocity to 3.0 0m/s 5. Set velocity to 4.0 0m/s 6. Set velocity to 10.00m /s Part 1 Directions: Before each run hit the button, set values and to make the man move Then DRAW the Position, Velocity and Acceleration graphs!
ASTM – Revision of ASTM B633 - Zinc Electroplating Standard . The IFI 2018 Annual report will detail that: IFI remains healthy and continues to build reserves, which remain over 2 million, which is sufficient for nearly two years of operations. Workforce development continues to be a major objective for the industry. With orders and production in the final months of 2018 .
