Lecture «Robot »: Intro to Dynamics

151-0851-00 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 1| 17.10.2017 | 1 19.09.2017 Intro and Outline Course Introduction; Recapitulation Position, Linear 26.09.2017 1 and ; Formulation, Transformation 26.09.2017 Exercise 1a Kinematics Modeling the ABB arm

03.10.2017 Kinematics 2 Kinematics of Systems of Bodies; Jacobians 03.10.2017 Exercise 1b Differential Kinematics of the ABB arm 10.10.2017 Kinematics 3 Kinematic Control Methods: Inverse Differential Kinematics, Inverse 10.10.2017 Exercise 1c Kinematic Control of the ABB Kinematics; Rotation Error; Multi-task Control Arm 17.10.2017 Dynamics L1 Multi-body Dynamics 17.10.2017 Exercise 2a Dynamic Modeling of the ABB Arm 24.10.2017 Dynamics L2 Floating Base Dynamics 24.10.2017 31.10.2017 Dynamics L3 Dynamic Model Based Control Methods 31.10.2017 Exercise 2b Dynamic Control Methods Applied to the ABB arm

07.11.2017 Legged Robot Dynamic Modeling of Legged Robots & Control 07.11.2017 Exercise 3 Legged robot 14.11.2017 Case Studies 1 Legged Robotics Case Study 14.11.2017 21.11.2017 Rotorcraft Dynamic Modeling of Rotorcraft & Control 21.11.2017 Exercise 4 Modeling and Control of Multicopter 28.11.2017 Case Studies 2 Rotor Craft Case Study 28.11.2017 05.12.2017 Fixed-wing Dynamic Modeling of Fixed-wing & Control 05.12.2017 Exercise 5 Fixed-wing Control and Simulation 12.12.2017 Case Studies 3 Fixed-wing Case Study (Solar-powered UAVs - AtlantikSolar, Vertical Take-off and Landing UAVs – Wingtra) 19.12.2017 Summery and Outlook Summery; Wrap-up; Exam Robot Dynamics - Dynamics 1| 17.10.2017 | 2 Recapitulation of Kinematics

. Kinematics = description of . Translations and . Various representations (Euler, quaternions, etc.) . Instantaneous/Differential kinematics . Jacobians and geometric Jacobians . Inverse kinematics and control . Floating base systems (unactuated base and contacts)

Robot Dynamics - Dynamics 1| 17.10.2017 | 3 Dynamics in Robotics

Robot Dynamics - Dynamics 1| 17.10.2017 | 4 Dynamics in Robotics

Robot Dynamics - Dynamics 1| 17.10.2017 | 5 Dynamics Outline T Mqq  bqq ,   gq  τ JFcc . Description of “cause of ” q Mq matrix . Input τ / acting on system  bqq, Centrifugal and Coriolis . Outputq Motion of the system  gq forces . Principle of virtual τ Generalized forces . Newton’s law for particles Fc External forces  Conservation of and angular Jc Contact Jacobian . 3 methods to get the EoM . Newton-Euler: Free cut and conservation of impulse & for each body . Projected Newton-Euler (generalized coordinates) . Lagrange II () . Introduction to dynamics of floating base systems . External forces

Robot Dynamics - Dynamics 1| 17.10.2017 | 6 Principle of Virtual Work

. Principle of virtual work (D’Alembert’s Principle) . Dynamic equilibrium imposes zero virtual work dFext external forces acting on element i r of element i dm mass of element i variational parameter r virtual of element i . Newton’s law for every particle in direction it can move

Fa m rFr m a r d  d  dm ΓrvN m Fvp m   r   dt N dt p   dF S force Impulse or angular momentum linear momentum Robot Dynamics - Dynamics 1| 17.10.2017 | 7 Virtual Displacements of Single Rigid Bodies

. Rigid body Kinematics

. Applied to principle of virtual work

Robot Dynamics - Dynamics 1| 17.10.2017 | 8 Impulse and angular momentum

. Use the following definitions

. Conservation of impulse and angular momentum Newton A free body can move In all directions Euler External forces and moments Change in impulse and angular momentum

Robot Dynamics - Dynamics 1| 17.10.2017 | 9 1st Method for EoM Newton-Euler for single bodies

. Cut all bodies free Fi2 . Introduction of constraining force Ω to individual bodies Ψ ۼ and ܘ Apply conservation .

mi ai

 System of equations vi F . 6n equation i1 . Eliminate all constrained forces (5n) r Fg OSi . Pros and Cons + Intuitively clear {I} + Direct access to constraining forces − Becomes a huge combinatorial problem for large MBS

Robot Dynamics - Dynamics 1| 17.10.2017 | 10 Free Cut Cart example

. Find the equation of motion

mcc, g {I} l

mp , p

Robot Dynamics - Dynamics 1| 17.10.2017 | 11 Newton-Euler in Generalized Motion Directions

. For multi-body systems . Express the impulse/angular momentum in generalized coordinates

. Virtual displacement in generalized coordinates

. With this, the principle of virtual work transforms to

0W=  qT q

Mq bqq,  Robot Dynamicsgq - Dynamics 1 | 17.10.2017| 15 Projected Newton-Euler

. Equation of motion Mqq  bqq ,   gq  0 . Directly get the dynamic properties of a multi-body system with n bodies

. For actuated systems, include actuation force as external force for each body . If actuators act in the direction of generalized coordinates, ࣎ corresponds to stacked actuator commands

Robot Dynamics - Dynamics 1| 17.10.2017 | 16 Projected Newton-Euler Cart pendulum example

. Find the equation of motion

mcc, g {I} l

mp , p

Robot Dynamics - Dynamics 1| 17.10.2017 | 17 3rd Method for EoM Lagrange II . Lagrangian . Lagrangian equation

inertial forces gravity vector 1  q  qMqT . Since UU d  T 2  qq,  TTU τ T T dt qqq

T M qq q1  1    with T  Mq MqM qqq  gg Mqq  b , q τ  2 q M  T  q q  qn 

Robot Dynamics - Dynamics 1| 17.10.2017 | 19 Lagrange II Kinetic energy 1  qMqT . Kinetic energy in joint T 2

. Kinetic energy for all bodies

. From kinematics we know that

. Hence we get

Robot Dynamics - Dynamics 1| 17.10.2017 | 20 Lagrange II Potential energy

. Two sources for potential forces

. Gravitational forces

rr 0 . Spring forces FrrEjkd00 rr 0

Robot Dynamics - Dynamics 1| 17.10.2017 | 21 Lagrange II Cart pendulum example

. Find the equation of motion

mcc, g {I} l

mp , p

Robot Dynamics - Dynamics 1| 17.10.2017 | 22 External Forces

. Given:

. Generalized forces are calculated as:

. Given:

. Generalized forces are calculated

. For actuator :

Robot Dynamics - Dynamics 1| 17.10.2017 | 24 External Forces Cart pendulum example

. Equation of motion without actuation x 2 {I} mm  lmcos    lm sin  0 m , cp p  p cc g 2 q  0 lmcos m l  0 mglsin  a ppp p   M b g l . Add actuator for the pendulum  a T   J  01 m , . Action on pendulum p a Rp TTT0  p p τJTJTJT Ri, i Rc c Rp p  T   J Rc  00  . Reaction on cart ca a l

Fx kx 2sin l   xs  . Add spring to the pendulum . (world attachment point P, zero length 0, stiffness k) Fkys2cos l  y . Action on pendulum xl 2sin  Pxys , s  r   2cosl   Fx   Fx   T F  τJFs   s    2cossinlF F  Fy r 12cosl   xy    J s   s   q 02sinl   Robot Dynamics - Dynamics 1| 17.10.2017 | 25 External Forces Cart pendulum example

. What is the external force coming from the motor

x {I} m , cc g M l

 mp , p l

Robot Dynamics - Dynamics 1| 17.10.2017 | 26 External Forces Cart pendulum example

. What is the external force coming from the motor

. Action on cart Fact x {I} m , T  Fact  cc g J  10 τJFc FFcact Pc  0    F act l

 mp , p l

Robot Dynamics - Dynamics 1| 17.10.2017 | 27