Jacobian: Velocities and Static Forces 1/4

Transcription

1 Jacobian: Velocities and Static Forces /4 Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

2 Kinematics Relations - Joint & Cartesian Spaces A robot is often used to manipulate object attached to its tip (end effector). The location of the robot tip may be specified using one of the following descriptions: Joint Space Cartesian Space R P N N NT Euler Angles N X P r N N {N} Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

3 Kinematics Relations - Forward & Inverse The robot kinematic equations relate the two description of the robot tip location N Tip Location in Joint Space X FK( ) IK(X ) X P r N N Tip Location in Cartesian Space Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

4 Kinematics Relations - Forward & Inverse N dt d ] [ z y z y N N v v v v X dt d X ] [ Tip Velocity in Joint Space Tip velocity in Cartesian Space Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

5 Jacobian Matri - Introduction The Jacobian is a multi dimensional form of the derivative. Suppose that for eample we have 6 functions, each of which is a function of 6 independent variables We may also use a vector notation to write these equations as ),,,,, ( ),,,,, ( ),,,,, ( f y f y f y Y F(X ) Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

6 Jacobian Matri - Introduction If we wish to calculate the differential of as a function of the differential we use the chain rule to get Which again might be written more simply using a vector notation as f f f y f f f y f f f y i y i X X F Y Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

7 Jacobian Matri - Introduction The 66 matri of partial derivative is defined as the Jacobian matri F Y X J ( X ) X X By dividing both sides by the differential time element, we can think of the Jacobian as mapping velocities in X to those in Y Y J( X ) X Note that the Jacobian is time varying linear transformation Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

8 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 by: N J J Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

9 Jacobian Matri - Introduction This epression can be epanded to: Where: is a 6 vector of the end effector linear and angular velocities is a 6N Jacobian matri is a N vector of the manipulator joint velocities is the number of joints N z y J z y N J N 6 6N N Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

10 Jacobian Matri - Introduction In addition to the velocity relationship, we are also interested in developing a relationship between the robot joint torques ( forces and moments ( F ) and the ) at the robot end effector (Static Conditions). This relationship is given by: F T J F Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

11 Jacobian Matri - Introduction This epression can be epanded to: Where: is a 6 vector of the robot joint torques is a 6N Transposed Jacobian matri is a N vector of the forces and moments at the robot end effector is the number of joints z y z y T N M M M F F F J F T J N 6 6N N Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

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15 Jacobian Matri - Calculation Methods Differentiation the Forward Kinematics Eqs. Iterative Propagation (Velocities or Forces / Torques) Jacobian Matri Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

16 Jacobian Matri by Differentiation - R - /4 Consider a simple planar R robot y P y r, P V ee The end effector position is given by P P y r cos y r sin Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

17 Jacobian Matri by Differentiation - R - /4 The velocity of the end effector is defined by V V y P P y r sin r sin y r cos r cos Epressed in matri form we have J y rsin r cos Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

18 Jacobian Matri by Differentiation - R - /4 y F ee F y P y r F, P The moment about the joint generated by the force acting on the end effector is given by rf sin rf y cos Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

19 Jacobian Matri by Differentiation - R - 4/4 Epressed in matri form we have T J F F F rsin r cos y J y rsin r cos Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

20 Jacobian Matri by Differanciation - R - /4 Consider the following DOF Planar manipulator y y y y y Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

21 Jacobian Matri by Differanciation - R - /4 Problem: Compute the Jacobian matri that describes the relationship J J T F Solution: The end effector position and orientation is defined in the base frame by y Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

22 Jacobian Matri by Differanciation - R - /4 The forward kinematics gives us relationship of the end effector to the joint angles: P org, L c L c L c P org, y P org, y L s L s L s Differentiating the three epressions gives L s L s y L c L s L s L s L s L s L s L s L c L c L c L c L c L c L c L c Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

23 Jacobian Matri by Differanciation - R - 4/4 Using a matri form we get J y L s Ls Ls Ls Ls Ls L c Lc Lc Lc L c Lc The Jacobian provides a linear transformation, giving a velocity map and a force map for a robot manipulator. For the simple eample above, the equations are trivial, but can easily become more complicated with robots that have additional degrees a freedom. Before tackling these problems, consider this brief review of linear algebra. Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

24 Singularity - The Concept Motivation: We would like the hand of a robot (end effecror) to move with a certain velocity vector in Cartesian space. Using linear transformation relating the joint velocity to the Cartesian velocity we could calculate the necessary joint rates at each instance along the path. J Given: a linear transformation relating the joint velocity to the Cartesian velocity (usually the end effector) Question: Is the Jacobian matri invertable? (Or) Is it nonsingular? Is the Jacobian invertable for all values of? If not, where is it not invertable? Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

25 Singularity - The Concept Answer (Conceptual): Most manipulator have values of where the Jacobian becomes singular. Such locations are called singularities of the mechanism or singularities for short Singularities of the mechanism Workspace boundary singularities Workspace interior Singularities - Stretched out - Folded back End Effector Workspace Boundary - Two or more joints are lining up Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

26 Singularity - The Concept Manipulator Singular Configuration General Configuration Losing One or More DOF All DOF Are Available Losing one or more DOF means that there is a some direction (or subspace) in Cartesian space along which it is impossible to move the hand of the robot (end effector) no matter which joint rate are selected Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

27 Singularity Physical Interpretation - Eamples Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

28 Brief Linear Algebra Review - / Inverse of Matri A eists if and only if the determinant of A is non-zero. A Eists if and only if Det( A) A If the determinant of A is equal to zero, then the matri A is a singular matri Det( A) A A Singular Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

29 Brief Linear Algebra Review - / The rank of the matri A is the size of the largest squared Matri S for which Det( S) Eample - A A S A S Rank ( A) Eample - A S S Rank ( A) Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

30 Brief Linear Algebra Review - / If two rows or columns of matri A are equal or related by a constant, then Eample ) ( A Det A ) det( A A Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

31 Brief Linear Algebra Review - 4/ Eigenvalues AX X ( AI ) X Eigenvalues are the roots of the polynomial X Det( AI) If each solution to the characteristic equation (Eigenvalue) has a corresponding Eigenvector Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

32 Brief Linear Algebra Review - 4/ A ) ( I A Det ) ( X X X I A Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

33 Brief Linear Algebra Review - 4/ X X X X X X Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

34 Brief Linear Algebra Review - 5/ Det( A) Any singular matri ( ) has at least one Eigenvalue equal to zero Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

35 Brief Linear Algebra Review - 6/ Det( A) If A is non-singular ( ), and is an eigenvalue of A with corresponding to eigenvector X, then A X X Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

36 Brief Linear Algebra Review - 7/ If the n n matri A is of full rank (that is, Rank (A) = n), then the only solution to AX is the trivial one X If A is of less than full rank (that is Rank (A) < n), then there are n-r linearly independent (orthogonal) solutions j j n r for which A j Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

37 Brief Linear Algebra Review - 8/ If A is square, then A and A T have the same eigenvalues Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

38 Properties of the Jacobian - Velocity Mapping and Singularities Eample: Planar R det( J ( )) L s L c L L c s L s L c L L c s L s Lc L s L c L L s det( J( )) L Ls det( J( )) Note that is not a function of, Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

39 Properties of the Jacobian - Velocity Mapping and Singularities singular configurat ion Stretched Out Fold Back The manipulator loses DEF. The end effector can only move along the tangent direction of the arm. Motion along the radial direction is not possible. Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

40 Properties of the Jacobian - Force Mapping and Singularities The relationship between joint torque and end effector force and moments is given by: T J F T J The rank of is equals the rank of. At a singular configuration there eists a non trivial force such that J J T F F In other words, a finite force can be applied to the end effector that produces no torque at the robot s joints. In the singular configuration, the manipulator can lock up. Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

41 Properties of the Jacobian - Force Mapping and Singularities Eample: Planar R ; F In this case the force acting on the end effector (relative to the {} frame) is given by F Fc Fs Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

42 Properties of the Jacobian - Force Mapping and Singularities For we get Fs Fc L c L s L c L c L s s L L c L c L c L s s L L s F J T ; ) ( ) ( ) ( ) ( ) ( ) ( L c Fs L c Fs L L c Fs L L c Fs L L L c Fs L L L c Fs Fs Fc L c L s L c L c L s s L L c L c L c L s s L L s F J T Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

43 Properties of the Jacobian - Force Mapping and Singularities This situation is an old and famous one in mechanical engineering. For eample, in the steam locomotive, top dead center refers to the following condition The piston force, F, cannot generate any torque around the drive wheel ais because the linkage is singular in the position shown. Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

44 Properties of the Jacobian - Velocity Mapping and Singularities We have shown the relationship between joint space velocity and end effector velocity, given by J It is interesting to determine the inverse of this relationship, namely J Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

45 Consider the square 66 case for. Properties of the Jacobian - Velocity Mapping and Singularities J Det J If rank < 6 ( ), then there is no solution to the inverse equation (see Brief Linear Algebra Review -,7). Rank J 6 J However, if the rank = 5, then there is at least one non-trivial solution to the forward equation (see Brief Linear Algebra Review - 7). That is, for J Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

46 Properties of the Jacobian - Velocity Mapping and Singularities The solution is a direction in the in joint velocity space for which joint motion produces no end effector motion. We call any joint configuration for which a singular configuration. Q Rank J 6 Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

47 Properties of the Jacobian - Velocity Mapping and Singularities For certain directions of end effector motion, i i 6 J i i where: i are the eigenvalues of i are the eigenvectors of J J J If is fully ranked (see Brief Linear Algebra Review - 6/ ), we have i J i Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

48 Properties of the Jacobian - Velocity Mapping and Singularities As the joint approach a singular configuration there is at least one eigenvalue for which i. This results in Q i i In other word, as the joints approach the singular configuration, the end effector motion in a particular task direction causes the joint velocities to approach infinity. However, there are task velocities that can have solutions. j J If loses rank by only one, then there are n- eigenvectors in the task velocity space ( solutions. j ) for which solutions do eist. However, there can be multiple Advanced Robotic - MAE 6D - Department of Mechanical & Aerospace Engineering - UCLA

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