Lecture «Robot Dynamics»: Dynamics and Control
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1 Lecture «Robot Dynamics»: Dynamics and Control 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 - Dynamics
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 - Dynamics
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 qb q q g S q M q T T q τ JF c c Generalized coordinates Mass matrix bqq, Centrifugal and Coriolis forces g q τ S F J c c Gravity forces Generalized fores c Selection matrix/jacobian External forces Contact Jacobian 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 FRANKA an example of a force controllable robot arm Robot Dynamics - Dynamics
10 ANYpulator An example for a robot that can interact Special force controllable actuators Dynamic motion Safe interaction position force motor gear spring link Series Elastic Actuator Robot Dynamics - Dynamics
11 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
12 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
13 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 for velocity control) Same priority, multi-task inversion Hierarchical Robot Dynamics - Dynamics
14 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
15 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
16 Robots in Interaction There is a long history in robots controlling motion and interaction Robot Dynamics - Dynamics
17 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
18 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
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 Inverse-dynamics Multiple solutions Robot Dynamics - Dynamics
25 Some Examples of Using Internal Forces Robot Dynamics - Dynamics
26 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
27 Internal Forces extreme example Robot Dynamics - Dynamics
28 Robot Dynamics - Dynamics
29 Least Square Optimization some notes on quadratic optimization Ax b 0 x=a b min Ax b min x 2 2 x x1 x1 Ax 1 1b Ax min 2 2 = A1 A2 b A1 A2 b x1, x2 x2 x 2 2 min x x Axb Axb Equal priority A1 b x= x x A 2 b min A b min x 2 A b Hierarchy x A b N A x N Axb A Ab A x b x A A b A A b min x min x st.. Axb Axb Axb c 0 2N Robot Dynamics - Dynamics
30 Least Square Optimization Application to Inverse Dynamics Mq b g τ Jq Jq w * e e e q M I bg 0 τ * J 0 Jq w e q τ e Single task min qτ, M Iq bg J 0 τ Jq w * e e q * min Je 0 Jq w e qτ, τ Priority 2 q st..m I bg0 τ 2 Robot Dynamics - Dynamics
31 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
32 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
33 Behavior as Multiple Tasks Robot Dynamics - Dynamics
34 Quasi-static: Virtual Model Control Pratt 2001 No dynamic effects Add virtual external forces to pull/support the robot Static equilibrium of forces and moments From principle of virtual work it follows directly that Robot Dynamics - Dynamics
35 Next Time Application of this technique for locomotion control of legged robots Robot Dynamics - Dynamics
36 Robot Dynamics - Dynamics
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