Robot Dynamics Instantaneous Kinematiccs and Jacobians

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1 Robot Dynamics Instantaneous Kinematiccs and Jacobians V Lecture: Tuesday 10:15 12:00 CAB G11 Exercise: Tuesday 14:15 16:00 every 2nd week Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny Marco Hutter Robot Dynamics - Kinematics

2 Kinematics Precise and fast position control Machines are built and controlled to achieve extremely accurate positions, independent of the load the robot carries Very stiff structure Play-free gears and transmissions High-accurate joint sensors End-effector accuracy +/- 0.02mm! Marco Hutter Robot Dynamics - Kinematics

3 Short Recap of Last Week Generalized coordinates, configuration (joint & operational space) Representation (incl. quaternions tutorial) Rotations are matrix operation Rotations can be differently represented (c.f. polar or Cartesian coordinates) Euler angles,, Rotation matrix Quaternions, Quaternions and rotational velocities 1 2 cos sin Rotation matrix 2 1 2, : Different conventions (JPL/Hamiltonian) Marco Hutter Robot Dynamics - Kinematics Different writing styles

4 Instantaneous Kinematics Forward Kinematics Instantaneous (or differential) Kinematics x x x x Jacobian has basis x x x f First order Taylor expansion f 2 xx f f O f J I J I x Jacobian Matrix {I} {E} Marco Hutter Robot Dynamics - Kinematics

5 Jacobians by direct differentiation Planar example, position Jacobian Express end-effector position as a function of generalized coordinates x l sin l sin l sin z l0 l1cos 1 l2cos 1 2 l3cos I Jacobian J x f lclc lc lc lc lc lsls ls ls ls ls This is often used for control inverse kinematics 2 nd hour today x J J x Marco Hutter Robot Dynamics - Kinematics

6 Jacobians by direct differentiation Rotation Jacobian End-effector configuration is position and orientation Parameterization of position Parameterization of rotation Planar rotations are simple x fp P x x R fr xr 12 3 J R Depending on the representation rotations are matrices there exits various parametrization, q0 q 1 e.g. Euler: Quaternions: x R xr q2 q3 x The Jacobian J is dependent on the representation R x, y, z IE I E I E I E Marco Hutter Robot Dynamics - Kinematics

7 Basic Jacobian Keep in mind Different representations of the end-effector position and orientation result in different kinematic models (and different Jacobian matrices). However, the kinematic properties are expected to be independent of the type of representation used for the description of the end-effector configuration. These properties are described by a basic kinematic model that is defined independently from the selected end-effector representation. This model relies on the end-effector linear and angular velocities. v E end-effector linear and angular velocity ve E 61 J ( ) 0 6n n1 Basic Jacobian J 0 this is unique!! {I} {E} E Marco Hutter Robot Dynamics - Kinematics

8 Jacobian Matrix Relation Jacobian associated with representation can be obtained from basic Jacobian EP ( x) 0 Jx( ) mn Ex( x) m6j0( ) 6n 0( ) R ( x) J 0 E v and are directly related to the representation E E x P EP xp ve x E x R R R E Marco Hutter Robot Dynamics - Kinematics

9 Some Examples Spherical coordinates E xp P xp ve x E x R R R E x rcos y rsin z z sin cos x rcos r y rsin r z z x P r z ysin cos xcos xsin r y r z cos sin 0 x sin r cos r 0 y z E P Marco Hutter Robot Dynamics - Kinematics

10 Some Examples Euler angles Parameterization,, Consecutive rotations Angular velocity Quaternions Lecture R R R 0 c s s s cs c R R R sc / s cc / s 1 x c s 0 s / s c / s 0 R I IE E x P EP xp ve x E x 0 c ss 0 s cs 1 0 c E 1 R R Marco Hutter Robot Dynamics - Kinematics R R R E

11 Basic Jacobian Given a representation x x x P R Representation dependent Jacobian x J x Basic Jacobian ve E ( ) J0 Goal: Find and as a function of generalized velocities Relation J E J 0 x x x Marco Hutter Robot Dynamics - Kinematics

12 Basic Rigid Body Kinematics Find the linear and angular velocity at the end-effector All vectors in same frame!! Linear velocity Adding velocities v v v v v v v v IE IF 1 FF 1 2 Fn E I IE I IF I FF I F E n {F 1 } {F n } {E} v E E Differentiation rule for vectors in moving frames d v r r r dt C AB C AB C AB C IC C AB {I} Marco Hutter Robot Dynamics - Kinematics

13 A Single Body Get velocity of P with respect to 0 expressed in I v v v v v R v v r r I OP I OB I BP R I OP I OB IB B BP I OB IB B BP B IB B BP Marco Hutter Robot Dynamics - Kinematics

14 Simple example linear velocity I vie? v E J P? Marco Hutter Robot Dynamics - Kinematics

15 Simple example linear velocity v v v v v I IE I IA I AB I BC I CE v v 0 0 d d r 0 0 dt dt l o 0 I IA I IA R v I AB IA A AB R r r IA A AB A IA A AB v R v R R r r I BC IB B BC IA AB B BC B IB B BC cos 0 sin l cos sin 1 0 cos10 0 l 1 1l1sin 1 Marco Hutter Robot Dynamics - Kinematics v E l cos 1 0 sin 1 cos 2 0 sin cos sin 1 0 cos1sin 2 0 cos20 0 l si l n 1 2

16 Simple example linear velocity cont I v CE l cos l3sin 12 3 v E v v v v v I IE I IA I AB I BC I CE 0 1l1cos l2cos l3cos l sin l sin l sin lc 1 1lc 2 12l3c123 lc 2 12 l3 c123 l3 c ls 1 1 l2 s12 l3 s123 l2 s12 l3 s123 l3 s123 3 J P Marco Hutter Robot Dynamics - Kinematics

17 Basic Rigid Body Kinematics Find the linear and angular velocity at the end-effector {E} E Linear velocity Adding velocities v v v v v v v v IE IF 1 FF 1 2 Fn E I IE I IF I FF I F E n {F 1 } {F n } v E Differentiation rule for vectors in moving frames d v r r r dt Angular velocity C AB C AB C AB C IC C AB Adding angular velocity I IE I IF I FF I F E n {I} Marco Hutter Robot Dynamics - Kinematics

18 Simple example angular velocity E I IE? J R? Marco Hutter Robot Dynamics - Kinematics

19 Simple example angular velocity E I IE I IA I AB I BC I CE R R R 0 I IE I IA IA A AB IA AB B BC 0 c1 0 s1 0 c1 0 s1c2 0 s s1 0 c1 0 s1 0 c1s2 0 c J R Marco Hutter Robot Dynamics - Kinematics

20 Exercise this afternoon! Implement this procedure for the ABB IWR120 Use Matlab to implement the algebraic routines v E {E} E {I} Marco Hutter Robot Dynamics - Kinematics

21 Robot Dynamics Inverse Kinematics V Lecture: Tuesday 10:15 12:00 CAB G11 Exercise: Tuesday 14:15 16:00 every 2nd week Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny Marco Hutter Robot Dynamics - Kinematics

22 Forward Kinematics Description of end-effector configuration (position & orientation) as a function of joint coordinates Transformation matrix T T T T T T T T IE Parametrized description x E x xr p f {I} {E} Configuration end-effector Forward Kinematics Joint angles Marco Hutter Robot Dynamics - Kinematics

23 Inverse Kinematics Description of end-effector configuration (position & orientation) as a function of joint coordinates Transformation matrix T T T T T T T T IE solve Parametrized description x E x T T 06 IE p 1 f x f x E r m equations n=6 unknows 12 equations n=6 unknows {I} {E} Configuration end-effector Inverse Kinematics Joint angles Marco Hutter Robot Dynamics - Kinematics

24 Closed Form Solutions Geometric or Algebra Analytic solutions exist for a large class of mechanism 3 intersecting neighboring axes (most industrial robots) Marco Hutter Robot Dynamics - Kinematics

25 Closed Form Solutions Geometric or Algebra Analytic solutions exist for a large class of mechanism 3 intersecting neighboring axes (most industrial robots) Geometric Decompose spatial geometry of manipulator into several plane problems and apply geometric laws Marco Hutter Robot Dynamics - Kinematics

26 Closed Form Solutions Geometric or Algebra Analytic solutions exist for a large class of mechanism 3 intersecting neighboring axes (most industrial robots) Geometric Algebraic Manipulate transformation matrix equation to get the angles TIE T01 1T12 2T233 T344 T455 T566 1 T T T T T T T 01 1 IE T01 1T122 TIE T233 T344T45 5T566 1 T01 1 T12 2 T23 3 TIE T34 4 T45 5 T56 6 Marco Hutter Robot Dynamics - Kinematics

27 Numerical Solution Differential Kinematics We have seen that x J use this to iteratively solve the problem 1. Start from initial guess (e.g. given by forward kin) 2. Evaluate the Jacobian x J 3. Invert the Jacobian to obtain i J x i, x i 4. Update generalized coordinates J x x i1 i goal i Marco Hutter Robot Dynamics - Kinematics

28 Inverse Kinematics 3-link arm example Determine end-effector Jacobian of a 3-Link Arm 1. Introduce coordinate frames 2. Introduce generalized coordinates q q q 3. Determine end-effector position l l cos q l cos q q l cos q q q 0rOF l1sin q1l2sin q1q2l3sin q1 q2 q Take the partial derivative 0rOF 0JOF l sin q l sin q q l sin q q q l sin q q l sin q q q l sin q q q l1cos q1 l2cos q1q2 l3cos q1 q2 q3 l2cos q1 q2 l3cos q1q2 q3 l3cos q1q2 q T Marco Hutter Robot Dynamics - Kinematics

29 Inverse Kinematics 3-link arm example Find desired configuration Apply inverse kinematics r q q F With zero start point 0 q0 0 0 Use iterative numerical solution q q J r r Start value i1 i goal i q Marco Hutter Robot Dynamics - Kinematics

30 Inverse Kinematics 3-link arm example Find desired configuration Apply inverse kinematics r q q F With zero start point 0 q q Use iterative numerical solution q q J r r i1 i goal i 0 Start value q0 0 0 Multiple solutions possible 2 dimensional problem but 3 joints Depending on start configuration 90 q0 0 0 Marco Hutter Robot Dynamics - Kinematics

31 Inverse Kinematics Iterative Methods Presented method = Newton method goal r r q r Jq qi 1 qi J r r qi What happens in singular positions? (target out of reach) J becomes singular Small can lead to large r q instable!!! r goal r q i 31

32 Inverse Kinematics Iterative Methods min J qr Presented method = Newton method goal r r q r Jq q q J r r q i1 i i 2 Damping (Levenberg-Marquardt) T T 2 qj JJ I r 1 min Jqr q Jacobi-transposed (steepest decent) Understand error as force pulling to the goal Jacobi-transposed mapping Linear scaling Possible choice for T q J r T T Introduction to Inverse Kinematics with rjj Jacobian r Transpose, Pseudoinverse and T T T Damped Least Squares methods, Samuel rjjjjbuss, r 2009 J T F 32

33 Differential Kinematics 3-link arm example Relation between Cartesian velocity and generalized velocities r J r F Inverse differential kinematics Given the end-effector velocity Determine the generalized velocities r J r Marco Hutter Robot Dynamics - Kinematics

34 Redundancy and Singularity 3-link arm example Jacobian is often not invertible => use the Moore-Penrose pseudo inverse J Redundancy Jacobian is column-rank deficient Pseudo inverse minimizes 2 Can have multiple solution goal J r IJ J 0 N goal r F Singularity Jacobian is row-rank deficient Pseudo inverse minimizes the error r goal r 2 goal r F r F Marco Hutter Robot Dynamics - Kinematics

35 Mapping Associated with Jacobian Marco Hutter Robot Dynamics - Kinematics

36 Robot Dynamics Static Forces V Lecture: Tuesday 10:15 12:00 CAB G11 Exercise: Tuesday 14:15 16:00 every 2nd week Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny Marco Hutter Robot Dynamics - Kinematics

37 Static Forces Velocity/Force duality 3 2 Principle of Virtual Work Variations in work must cancel for all virtual displacement Internal forces of ideal joint don t contribute Using T T i i E E 0 i W f x F x x J F E 1 T T J F E J 0 T F E Marco Hutter Robot Dynamics - Kinematics

38 Static Forces Velocity/Force duality x J J T F Marco Hutter Robot Dynamics - Kinematics

39 Static Forces Example What are the joint torques for unitary link lengths and a load of Fload 1N joint angles of 1 0, 2 60 c1 c12 r s1 s F s s s J c c c s1s12 s12 0 c1 c12 c T 1 2 Marco Hutter Robot Dynamics - Kinematics

40 Robot Dynamics Floating Base Systems V Lecture: Tuesday 10:15 12:00 CAB G11 Exercise: Tuesday 14:15 16:00 every 2nd week Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny Marco Hutter Robot Dynamics - Kinematics

41 Kinematic Structure of Mobile Robots Legged robot r Contact constraints Footpoint is not allowed to move r F J 0 F b Chose the joint speed such that this constraint is satisfied 0 JFr F N 0 F Generalized coordinates b r Un-actuated base Actuated joints Move body upward r J J N body body body F 0 0 JFr F N F0 NF JbodyNF r body Marco Hutter Robot Dynamics - Kinematics

42 Kinematic Structure of Mobile Robots Wheeled platform Contact constraints Point on wheel Jacobian I I x rsin rop rrcos 0 J P 1 r cos 0 r sin 0 0 Generalized coordinates x Un-actuated base Actuated joints Contact constraints x I P I P => Rolling condition 1 r x J x r 0 Marco Hutter Robot Dynamics - Kinematics

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