Lecture «Robot Dynamics»: Dynamics 2

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1 Lecture «Robot Dynamics»: Dynamics 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 - Dynamics

2 Topic Title Intro and Outline L1 Course Introduction; Recapitulation Position, Linear Velocity, Transformation Kinematics 1 L2 Rotation Representation; Introduction to Multi-body Kinematics Exercise 1a E1a Kinematics Modeling the ABB arm Kinematics 2 L3 Kinematics of Systems of Bodies; Jacobians Exercise 1b L3 Differential Kinematics and Jacobians of the ABB Arm Kinematics 3 L4 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation Error; Multi-task Control Exercise 1c E1b Kinematic Control of the ABB Arm Dynamics L1 L5 Multi-body Dynamics Exercise 2a E2a Dynamic Modeling of the ABB Arm Dynamics L2 L6 Dynamic Model Based Control Methods Exercise 2b E2b Dynamic Control Methods Applied to the ABB arm Legged Robots L7 Case Study and Application of Control Methods Rotorcraft 1 L8 Dynamic Modeling of Rotorcraft I Rotorcraft 2 L9 Dynamic Modeling of Rotorcraft II & Control Exercise 3 E3 Modeling and Control of Multicopter Case Studies 2 L10 Rotor Craft Case Study Fixed-wing 1 L11 Flight Dynamics; Basics of Aerodynamics; Modeling of Fixed-wing Aircraft Exercise 4 E4 Aircraft Aerodynamics / Flight performance / Model derivation Fixed-wing 2 L12 Stability, Control and Derivation of a Dynamic Model Exercise 5 E5 Fixed-wing Control and Simulation Case Studies 3 L13 Fixed-wing Case Study Summery and Outlook L14 Summery; Wrap-up; Exam Robot Dynamics - Dynamics 2 2

3 Recapitulation We learned how to get the equation of motion in joint space Newton-Euler Projected Newton-Euler Lagrange II Introduction to floating base systems Today: How can we use this information in order to control the robot, M q q b q q g q J F τ q M q T c Generalized accelerations Mass matrix b q, q Centrifugal and Coriolis forces g q τ F J c c Gravity forces Generalized forces External forces Contact Jacobian c Robot Dynamics - Dynamics

4 Position vs. Torque Controlled Robot Arms Robot Dynamics - Dynamics

5 Setup of a Robot Arm System Control Actuator Control ( U, I) τ qqq,, Actuator (Motor + Gear) Robot Dynamics Robot Dynamics - Dynamics

6 Classical Position Control of a Robot Arm q, q High Actuator gain Control PID ( U, I) Actuator (Motor + Gear) τ Robot Robot Dynamics qqq,, Position Sensor Position feedback loop on joint level Classical, position controlled robots don t care about dynamics High-gain PID guarantees good joint level tracking Disturbances (load, etc) are compensated by PID => interaction force can only be controlled with compliant surface Robot Dynamics - Dynamics

7 Joint Torque Control of a Robot Arm q, q τ Actuator Control ( U, I) Actuator (Motor + Gear) τ Robot Dynamics qqq,, Torque Sensor Position Sensor Integrate force-feedback Active regulation of system dynamics Model-based load compensation Interaction force control Robot Dynamics - Dynamics

8 Setup of Modern Robot Arms Modern robots have force sensors Dynamic control Interaction control Safety for collaboration Robot Dynamics - Dynamics

9 ANYpulator An example for a robot that can interact Special force controllable actuators Dynamic motion Safe interaction Robot Dynamics - Dynamics

10 Joint Impedance Control Torque as function of position and velocity error Closed loop behavior Static offset due to gravity Impedance control and gravity compensation Estimated gravity term Robot Dynamics - Dynamics

11 Inverse Dynamics Control Compensate for system dynamics In case of no modeling errors, the desired dynamics can be perfectly prescribed PD-control law Every joint behaves like a decoupled mass-spring-damper with unitary mass Robot Dynamics - Dynamics

12 Inverse Dynamics Control with Multiple Tasks Motion in joint space is often hard to describe => use task space A single task can be written as In complex machines, we want to fulfill multiple tasks (As introduced already in for velocity control) Same priority, multi-task inversion Hierarchical Robot Dynamics - Dynamics

13 Task Space Dynamics Joint-space dynamics End-effector dynamics w e F e {E} Torque to force mapping Inertia-ellipsoid Kinematic relation Substitute acceleration {I} Robot Dynamics - Dynamics

14 End-effector Motion Control Determine a desired end-effector acceleration Note: a rotational error can be related to differenced in representation by Determine the corresponding joint torque E R χ R χ R w t e Trajectory control w e F e {E} {I} Robot Dynamics - Dynamics

15 Robots in Interaction There is a long history in robots controlling motion and interaction Robot Dynamics - Dynamics

16 Operational Space Control Generalized framework to control motion and force Extend end-effector dynamics in contact with contact force F c Introduce selection matrices to separate motion force directions w e F e F c {I} Robot Dynamics - Dynamics

17 Operational Space Control 2-link example Given: Find, s.t. the end-effector r accelerates with exerts the contact force T edes, 0 a y, 0 T contact des Fc F 1 y O l x l r edes, 0 a y P 2 F contact, des F 0 c Robot Dynamics - Dynamics

18 Operational Space Control 2-link example Given: Find, s.t. the end-effector r accelerates with exerts the contact force End-effector position and Jacobian ls1 ls12 lc lc lc re lc1lc J e 12 ls ls ls Desired end-effector dynamics ˆ T τ J Λ ˆ r μˆ pˆ F e T edes, 0 a y, 0 T contact des Fc F edes, contace, des Λ 1 T JM 1 e Je 1 μ ΛJM e bλjq 1 p ΛJeM g 1 y O l x l P 2 F r edes, contact, des 0 a y F 0 c Robot Dynamics - Dynamics

19 How to Find a Selection Matrix Selection matrix in local frame 1: it can move 0: it can apply a force Rotation between contact force and world frame Robot Dynamics - Dynamics

20 How to Find a Selection Matrix Selection matrix in local frame Rotation between contact force and world frame Robot Dynamics - Dynamics

21 Sliding a Prismatic Object Along a Surface Assume friction less contact surface Σ Mp Σ Fp Σ Σ Mr Σ Fr Robot Dynamics - Dynamics

22 Inserting a Cylindrical Peg in a Hole Find the selection matrix (in local frame) Σ Mp Σ Fp Σ Σ Mr Σ Fr Robot Dynamics - Dynamics

23 Inverse Dynamics of Floating Base Systems Robot Dynamics - Dynamics

24 Recapitulation: Support Consistent Dynamics Equation of motion (1) Cannot directly be used for control due to the occurrence of contact forces Contact constraint Contact force Back-substitute in (1), replace Jq s Jq s and use support null-space projection Support consistent dynamics Robot Dynamics - Dynamics

25 Inverse Dynamics of Floating Base Systems Equation of motion of floating base systems Support-consistent Inverse-dynamics Multiple solutions Robot Dynamics - Dynamics

26 Some Examples of Using Internal Forces Robot Dynamics - Dynamics

27 Recapitulation: 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 - Dynamics

28 Some Examples of Using Internal Forces Robot Dynamics - Dynamics

29 Operational Space Control as Quadratic Program A general problem We search for a solution that fulfills the equation of motion Motion tasks: Force tasks: Torque min: min τ 2 Robot Dynamics - Dynamics

30 Solving a Set of QPs QPs need different priority!! Exploit Null-space of tasks with higher priority Every step = quadratic problem with constraints Use iterative null-space projection (formula in script) Robot Dynamics - Dynamics

31 Behavior as Multiple Tasks Robot Dynamics - Dynamics

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