Lecture «Robot Dynamics»: Kinematics 3

Transcription

1 Lecture «Robot Dynamics»: Kinematics V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) Marco Hutter, Roland Siegwart, and Thomas Stastny Robot Dynamics - Kinematics

2 Intro and Outline Course Introduction; Recapitulation Position, Linear Velocity Kinematics 1 Rotation and Angular Velocity; Rigid Body Formulation, Transformation Exercise 1a Kinematics Modeling the ABB arm Kinematics 2 Kinematics of Systems of Bodies; Jacobians Exercise 1b Differential Kinematics of the ABB arm Kinematics 3 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation Error; Multi-task Control Exercise 1c Kinematic Control of the ABB Arm Dynamics L1 Multi-body Dynamics Exercise 2a Dynamic Modeling of the ABB Arm Dynamics L2 Floating Base Dynamics Dynamics L3 Dynamic Model Based Control Methods Exercise 2b Dynamic Control Methods Applied to the ABB arm Legged Robot Dynamic Modeling of Legged Robots & Control Exercise 3 Legged robot Case Studies 1 Legged Robotics Case Study Rotorcraft Dynamic Modeling of Rotorcraft & Control Exercise 4 Modeling and Control of Multicopter Case Studies 2 Rotor Craft Case Study Fixed-wing Dynamic Modeling of Fixed-wing & Control Exercise 5 Fixed-wing Control and Simulation Case Studies 3 Fixed-wing Case Study (Solar-powered UAVs - AtlantikSolar, Vertical Take-off and Landing UAVs Wingtra) Summery and Outlook Summery; Wrap-up; Exam Robot Dynamics - Kinematics

3 Multi-body Kinematics Intro 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! Large variety of robot arms Robot Dynamics - Kinematics

4 Fixed Base vs. Floating Base Robot Base frame is rigidly connected to ground Often indicated as CS 0 Base frame is free floating Often indicated as CS B (base) 6 unactuated DOFs! Robot Dynamics - Kinematics

5 Classical Serial Kinematic Linkages Generalized robot arm joints revolute (1DOF) prismatic (1DOF) 1links moving links 1 fixed link Robot Dynamics - Kinematics

6 Configuration Parameters Generalized coordinates 5 constraints 6 parameters 3 positions 3 orientations n moving links: 6n parameters n 1DoF joints: 5n parameters 6n 5n = n DoFs Generalized coordinates A set of scalar parameters q that describe the robot s configuration Must be complete (Must be independent) => minimal coordinates Is not unique Degrees of Freedom Nr of minimal coordinates Robot Dynamics - Kinematics

7 End-effector Configuration Parameters End-effector configuration parameters A set of m parameters that completely specify the end-effector position and orientation with respect to I {E} Operational space coordinates the m 0 configuration parameters are independent => m 0 number of degrees of freedom of end-effector {I} Robot Dynamics - Kinematics

8 End-effector Configuration Parameters Example Most general robot arm: q = = = = = SCARA robot arm q = = = = = ANYpulator: robot arm with 4 rotational joints q = = = = = Robot Dynamics - Kinematics

9 End-effector Configuration Parameters Example Most general robot arm: q = q... 1 q n j = = =,,, x, y, z = q =,,, = 6 = 4 = xyz,,, x, y, z = xyz,,, z ANYpulator: robot arm with 4 rotational joints q = q1, q2, q3, q4 = = = xyz,,, x, y, z = 6 6 xyz xyz,,, x, y, z SCARA robot arm 6 4 Robot Dynamics - Kinematics

10 End-effector Configuration Parameters Example Most general robot arm: q = q... 1 q n j = = =,,, x, y, z = q =,,, = 6 = 4 = xyz,,, x, y, z = xyz,,, z ANYpulator: robot arm with 4 rotational joints q = q1, q2, q3, q4 = = = xyz,,, x, y, z = 6 6 xyz xyz,,, x, y, z SCARA robot arm 6 4 Robot Dynamics - Kinematics

11 End-effector Configuration Parameters Simple example Planar robot arm 3 revolute joints 1 end-effector (gripper) <= don t consider this for the moment What are the joint coordinates (generalized coordinates)? What are the end-effector parameters? Robot Dynamics - Kinematics

12 Configuration Space Joint Space Joint Coordinates Operational Coordinates Robot Dynamics - Kinematics

13 Configuration Space Joint Space Joint Coordinates => Joint Space 3 1 q obstacle Operational Coordinates => Operational Space 1 z χ e x z x Robot Dynamics - Kinematics

14 Forward Kinematics End-effector configuration as a function of generalized coordinates n e For multi-body system, use transformation matrices Robot Dynamics - Kinematics

15 Forward Kinematics Simple example What is the end-effector configuration as a function of generalized coordinates? T T T T T T IE I E c1 0 s1 0 c2 0 s2 0 c3 0 s s1 0 c1 l0 s2 0 c2 l1 s3 0 c3 l l c123 0 s123 l1s1 l2s12 l3s s123 0 c123 l0 lc 1 1 l2c12 l3c Robot Dynamics - Kinematics

16 Forward Differential Kinematics Analytical Jacobian Forward Kinematics e P χ χ q e χ χ e R e Forward Differential Kinematics Robot Dynamics - Kinematics

17 Forward Differential Kinematics Analytical Jacobian Forward Kinematics e P χ χ q e χ χ e R e Forward Differential Kinematics Analytic: with Robot Dynamics - Kinematics

18 Analytical Jacobian Planar robot arm Given (from last example) Determine the analytical Jacobian Robot Dynamics - Kinematics

19 Analytical Jacobian Planar robot arm Given (from last example) Determine the analytical Jacobian Robot Dynamics - Kinematics

20 Forward Differential Kinematics Robot Dynamics - Kinematics

21 Forward Differential Kinematics Analytic: with Depending on parameterization!! Geometric: Independent of parameterization Algebra: Robot Dynamics - Kinematics

22 Velocity in Moving Bodies Definitions Remember the difference: Velocity Time derivative of coordinates: Robot Dynamics - Kinematics

23 Vector Differentiation in Moving Frame Euler differentiation rule Robot Dynamics - Kinematics

24 Vector Differentiation in Moving Frame Euler differentiation rule For non-moving reference frames: For moving reference frames: Vector differentiation in moving frames (A = inertial/reference frame): Robot Dynamics - Kinematics

25 Velocity in Moving Bodies Rigid body formulation Apply transformation rule as learned before Differentiate with respect to time Substitute Rigid body formulation Robot Dynamics - Kinematics

26 Velocity in Moving Bodies Rigid body formulation Apply transformation rule as learned before Differentiate with respect to time Substitute Rigid body formulation Robot Dynamics - Kinematics

27 Geometric Jacobian Derivation Rigid body formulation at a single element Apply this to all bodies Angular velocity propagation get the end-effector velocity Robot Dynamics - Kinematics

28 Geometric Jacobian Derivation Rigid body formulation at a single element Apply this to all bodies Angular velocity propagation with get the end-effector velocity Robot Dynamics - Kinematics

29 Geometric Jacobian Derivation Position Jacobian Rotation Jacobian from Robot Dynamics - Kinematics

30 Geometric Jacobian Planar robot arm Preparation: determine the rotation matrices Determine the rotation axes Locally Inertial frame Determine the position vectors = Get the Jacobian Robot Dynamics - Kinematics

31 Recapitulation Analytical and Kinematic Jacobian Analytical Jacobian Geometric (or basic) Jacobian Relates time-derivatives of config. parameters to generalized velocities Depending on selected parameterization (mainly rotation) in 3D χ q Note: there exist no rotation angle Mainly used for numeric algorithms Relates end-effector velocity to generalized velocities Unique for every robot Used in most cases Robot Dynamics - Kinematics

32 Importance of Jacobian Kinematics (mapping of changes from joint to task space) Inverse kinematics control Resolve redundancy problems Express contact constraints Statics (and later also dynamics) Principle of virtual work Variations in work must cancel for all virtual displacement Internal forces of ideal joint don t contribute i T T i i E E W fx τ q F x Dual problem from principle of virtual work T 0 T E τ q F Jq q x τ Jq J T F 3 2 F E Robot Dynamics - Kinematics

33 Floating Base Kinematics V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) office hour: LEE H303 Friday Marco Hutter, Roland Siegwart, and Thomas Stastny Robot Dynamics - Kinematics

34 Floating Base Systems Kinematics Generalized coordinates with Generalized velocities and accelerations? qq, Time derivatives depend on parameterization Often Linear mapping Robot Dynamics - Kinematics

35 Floating Base Systems Differential kinematics Position of an arbitrary point on the robot Velocity of this point I r IB q b CIB q b B r BQ q j ω C ω B C IB T IB C C BI C I IB C T T IB IB IB IB C C T BI IB I J Q q with Robot Dynamics - Kinematics

36 Contact Constraints A contact point is not allowed to move: Constraint as a function of generalized coordinates: Stack of constraints Robot Dynamics - Kinematics

37 Contact Constraints A contact point is not allowed to move: Constraint as a function of generalized coordinates: Stack of constraints Robot Dynamics - Kinematics

38 Contact Constraint Wheeled vehicle simple example Contact constraints Point on wheel Jacobian Contact constraints Robot Dynamics - Kinematics

39 Contact Constraint Wheeled vehicle simple example Contact constraints Point on wheel Jacobian I r J I IP P x rsin rrcos 0 1 r cos 0 rsin 0 0 Contact constraints I 1 r x IP P 0 0 r J q I => Rolling condition x r 0 q x Un-actuated base Actuated joints Robot Dynamics - Kinematics

40 Properties of Contact Jacobian Contact Jacobian tells us, how a system can move. Separate stacked Jacobian Base is fully controllable if relation between base motion and constraints n n n c b j Nr of kinematic constraints for joint actuators: - Generalized coordinates DON T correspond to the degrees of freedom Contact constraints! Minimal coordinates (= correspond to degrees of freedom) Require to switch the set of coordinates depending on contact state (=> never used) Robot Dynamics - Kinematics

41 Quadrupedal Robot with Point Feet Floating base system with 12 actuated joint and 6 base coordinates (18DoF) Total constraints Internal constraints Uncontrollable DoFs Robot Dynamics - Kinematics

42 Quadrupedal Robot with Point Feet Floating base system with 12 actuated joint and 6 base coordinates (18DoF) Total constraints Internal constraints Uncontrollable DoFs Robot Dynamics - Kinematics

43 Outlook Exercise TOMORROW Differential Kinematics Use it as extended office hour! Next Lecture Script Section 2.9 (Kinematic Control Methods) Inverse Kinematics Inverse Differential Kinematics {I} {E} Configuration end-effector Inverse Kinematics Joint angles Robot Dynamics - Kinematics