Approach based on Minimal coordinates · acceleration (rotational and translational) of each body...

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GraSMech – Multibody 1

Approach based on

Minimal coordinates

Prof. O. Verlinden

Université de Mons - Faculté polytechnique

[email protected]

GraSMech course 2009-2010

Computer-aided analysis of rigid

and flexible multibody systems


Modelling steps

� Choose the configuration parameters (q)

� Set up the kinematics: express position, velocity and

acceleration (rotational and translational) of each

body in terms of q and its first and second time

derivatives

� Express the forces in terms of q, its time derivatives

and time t

� Build the differential equations of motion

� Numerical treatment of the equations

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Principle of minimal coordinates

� Type of coordinates

(relative, absolute,…)

chosen freely as far as

number of coordinates=

number of degrees

of freedom

� The kinematics needs a dedicated work for each system

� not adapted to general software (not systematic)

� leads to ordinary differential equations (ODE), interesting

for control


Question

�Which choice is appropriate or not ?

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Answer

� Choices 1 and 6: OK

� Choices 2 and 3: not recommended as several positions possible for a given set of configuration parameters

� Choices 4 and 5: OOPS !

�Parameters not independent

�No configuration parameter for second body in choice 4


Question

For the double pendulum, what would be the number of equations of motion with

�Minimal coordinates ?

� Relative coordinates ?

� Cartesian coordinates (planar case) ?

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Answer

�Minimal coordinates: 2 (ODE)

� Relative coordinates: 2 ODE)

� Cartesian coordinates: 6+4=10 DAE

In 3D, it would be: 12+10=22


Modelling steps





derivatives


and time t



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The need for a geometrical formalism

Tool needed to express position and orientation of a body, through a coordinate system (or frame) attached to the body

global reference

frame

body

reference

frame

(center of

mass)

secondary

body

frame

For body i -> homogeneous transformation matrix T0,i


Vectors and coordinate systems

Tool needed to manipulate vectors (velocities, forces,

accelerations,...) in different coordinate systems

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Position vector of a point


Homogeneous transformation matrices

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Examples of matrices

� Displacement without rotation

� Rotation about X axis


Examples of matrices

� Rotation about Z axis

� Rotation about a unit vector n

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Switching between coordinate systems

Postion vectors relative to different coordinate systems

are linked by transformation matrices

The position of a point P of body i is easily

expressed from its local coordinates and T0,i


1) A rotation about Y is represented by

2) Whatever P and i

3) Whatever P, i and j

4) Whatever i and j

Right or wrong ?

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Answers

1) No

Wrong matrix: columns are not unitary and not

perpendicular to each other

2) Yes

3) Yes

4) Yes

Rotation matrices are orthogonal


Homogeneous transformation matrices are particularly interesting along kinematic chains (robotics)

Multiplication of position matrices

A complex matrix is built from the multiplication of

elementary matrices (rotation or translation)

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Question


Answer

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Position, velocity and parameters

Situation of each body expressed in terms of q

General form of velocity


Kinematic matrices

� Velocities can be expressed in matrix form

� as well as accelerations

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Example: double pendulum



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Global kinematics

� Kinematics of body i

� Kinematics of secondary frames

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Quizz

If q is the vector of configuration parameters chosen

according to the minimal coordinates approach, the

velocity of a point P in the system is of the form


Answer

1) Yes for scleronomic systems

(Constraints independent of time)

2) Yes for rheonomic systems

(Constraints are time dependent: a motion is

imposed somewhere)

3) Forget it !

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Modelling steps





derivatives


and time t




Expression of forces

For each body i, we need

� The resultant force of all applied forces exerted on

body i

� The resultant moment of all applied forces exerted on

body i, with respect to center of mass

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Force elements - examples

� Spring

(stiffness k and

rest length l0)

� Damper (damping

coefficient c)



The only applied force is the

gravity

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Question: element force or not ?

� A spring

� A damper

� An undeformable wheel rolling on the ground without

sliding

� A contact force between two undeformable surfaces

subjected to Coulomb’s conditions

� A motor torque defined as a time function

� A motor torque deriving from a proportional derivative

controller (on position)


Answers

� Spring: Yes (force depends on relative position)

� Damper: Yes (force depends on relative position and

velocity)

�Wheel: No (imposes kinematic constraints => joint)

� Contact force between undeformable bodies: No

kinematic constraint for the normal component

kinematic constraint tangentially if adhesion

force tangentially if sliding

�Motor torque: Yes (force function of time)

�Motor torque control: Yes (force function of error and

its derivatives)

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Modelling steps





derivatives


and time t




Equations of motion

How to get (in a systematic way now) the equations of

motion from

�Kinematics

�Applied efforts on each body

The theorem of virtual power (d’Alembert) is particularly

well suited

for any virtual motion and even

for any admissible virtual motion (principle of VP)

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Virtual power of applied forces

� Kinematically admissible motion

as kinematic step includes all the joints

� Power developed by applied forces


Virtual power of inertia forces

� Power developed by inertia forces

with mi and ΦGi the mass and tensor matrix of body i

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Equations of motion

Globally, the expression

becomes


Equations of motion

� Equations of motion=second-order ordinary differential equations (ODE)

� with the mass matrix

� the Coriolis and centrifugal terms

� and the contribution of applied forces

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Equivalent vector form

Equations of motion can also be obtained from partial

velocities

with for example for the mass matrix


Applicability of minimal coordinates

Equations of motion can be written from kinematics and

applied efforts !

But ... is the method applicable in practice ?

Yes -> Team of Prof. Manfred Hiller (University of

Duisburg, Germany), A. Keczkemethy,

M. Anantharaman

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Practical implementation

� Classical organization of a MBS software

Application 1

Generals

olver

Results 1

Application 2

Application 3

Results 2

Results 3

Data

files

� Organization with minimal coordinates

General

library

Results 1Dedicated program 1

Results 2

Results 3

Dedicated program 2

Dedicated program 3


Components of the library

� Routines to express motion (composition of

movements along a kinematic chain)

� Routines to solve constraints

� Routines to express forces (springs, dampers, tires,

contact,…)

� Routines to build equations of motion from kinematics

and applied forces

� Routines for the numerical treatment of equations of

motion (integration, static equilibrium, linearization,

computation of poles,…)

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Principle for generalization (Hiller)

Inside a loop, the motion is described in terms of

minimal coordinates (loop=kinematic transformer)

Independent configuration parameters

Dependent conf. parameters

Expression of motion

Solving of constraints

Basic ideas: constraints solved in advance and by loops


Generalization - example

Example: slider-crank mechanism

And so on for velocities and accelerations !!

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Kinematic transformer (Hiller)

Loop=kinematic transformer

Different strategies to solve loops: typical pair of joints,

minimal polynomial equation, automatic gneration of

closed-form solutions


Industrial example

Not limited to simplistic systems !

30 bodies

40 joints

13 loops

25 dof

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Other implementation: EasyDyn


Components of EasyDyn

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Using EasyDyn (C++ only)

The user must a provide a C++ program with

� Routine SetInertiaData() specifying the inertia properties of all bodies

� Routine ComputeMotion() implementing

� Routine AddAppliedEfforts() describing the efforts on each body (eventually with the help of available routines)

� A main routine for initialization and call of integration routines

The user can rely on the vec module for kinematics and efforts


CAGeM

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Using EasyDyn with CAGeM

Mupad script with

� Dimensions

� Inertia properties

� Kinematics (position)

� Gravity

� Initial conditions

� options

Core C++ program

� Kinematics (velocity

and acceleration)

� External forces

(gravity only)

� Typical simulation

(time integration)

CAGeM

Body i: TOG[i]=Trotz(q[0])*Tdisp(q[1],0,0)…

q[j]= configuration parameter

Number of configuration parameters=number of degrees

of freedom (minimal coordinates- no constraints)


Using EasyDyn with CAGeM (2)

Core C++ program

� Kinematics (velocity

and acceleration)

� External forces

(gravity only)

� Time integration

Add other forces

� Available routines

(spings, dampers,…)

� Vector library

Add other differential

equations (electrical,

pneumatic,

hydraulic,…)

Adapt simulation (static

equilibrium,

poles,discrete-time

operations)

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Double pendulum in MuPAD


Double pendulum in MupAD

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Some interesting features of CAGeM


Slider-crank mechanism

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Robot

3D doesn’t make any difference !


Interest of CAGeM

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Efforts of actuators

vectors

action

reaction


Walking robot AMRU5

AMRU: Autonomy of Mobile Robots in Unstructued

environments

AMRU5: hexapod robot built by RMA, ULB and VUB

•18 degrees of freedom

•hierarchical control:

central controller

+one controller per leg

(Microchip PIC)

•about 30 kg

•Diameter about 1 m

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Leg of AMRU5

Mechanism of the leg=pantograph

Horizontal and vertical motions of the

foot are driven independently

-> gravitationally decoupled actuation


Mechanics of the leg

Nut-screw

Motor-gearbox-encoder

Axis of rotation

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Model of AMRU5

� Contact with the ground introduced as external forces

� Total of 24 degrees of freedom

T0G[0]=Tdisp(q[0],q[1],q[2])

*Trotx(q[3])*Troty(q[4])*Trotz(q[5])


Kinematics of the pantograph

The situation of each body must be expressed from the 2

configuration parameters

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Kinematics of the pantograph


Summary of the model

� 49 bodies

� The central body

� 8 bodies per leg (support part, 4 leg members and 3 rotors)

� 24 configuration parameters(and second-order differentialequations) for the mechanical part

� PID Controllers

� 36 differential equations in continuous time (test only)

� Algebraic expressions in discrete time (actual controller)

� First-order model of DC motors

� Contact force derived from relative motion with respect to the ground (nonlinear elastic with damping)

Input of the simulation: targets of the controllers

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Interest of the model

The main interest of the model is to improve the gait

generation => softer and less energy consumption


Let’s have it walk !

Bad gait Good gait

Climbing

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Example: human walking

Interest of EasyDyn in Biomechanics

� Particular kinematics (not classical joints)

� Particular forces

�Contact forces (foot-ground)

�Resistive joint torques

�Muscle modelling

Rustin Cédric, F.S.R.-FNRS PhD student

University of Mons (UMONS)

7000 Mons (Belgium)


Human lower limbs and walking process

� Kinematics : Delp's model

� 14 bodies :

− Torso-head

− Pelvis

− 2 femurs

− 2 patellas

− 2 tibias-fibulas

− 2 talus

− 2 calcaneus

− 2 sets of toes

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� Kinematics : Delp's model

� 11 joints and 23 dof :− 6 for the pelvis

− 3 orientation angles for the torso-head

wrt the pelvis

− 3 orientation angles for each thigh

wrt the pelvis

− the knee angle for each calf

wrt each thigh

− the ankle angle for each talus

wrt each calf

− the angle for each calcaneus

wrt each talus

− the angle for each set of toes

wrt to each calcaneus



Human articulations are not classical

mechanical joints

For instance, for the femur-(tibia-fibula)

joint (one dof joint)TrefG[R_TIBIAFIBULA]:=

…

*Tdisp(g5_(q[R_KNEE_ang]),

g6_(q[R_KNEE_ang]),

R_tibia_fibula_kinematicstranslation_z)

*Trotz(q[R_KNEE_ang])

*…

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Passive joint moments: results form

the resistance and damping

generated by the skin, soft

tissues, cartilages, muscles and

tendons wrapping the joint.

Amankwah's model: passive

moment described by a

differential equation



Muscles (particular actuator)

� 88 muscles for the 2 lower limbs

� Can be modelled with the virtual

muscle Brown and Loeb’s models

6 ODE's per muscle

� the excitation and activation

processes

� the F-L and F-V filters

� the properties of the slow and

fast fiber types

� the yield and sag phenomena

� ...

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Foot-ground contacts: 6 ellipsoids of contact

Non linear contact force expressed in terms of penetration

and penetration rate


Present work: dynamic simulation of walking

Presently input data = joint angles and velocities

Soon (hopefully) input data = muscles activation history

Input data must be adapted by optimzation for a stable

walking

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Conclusion

� The approach based on minimal coordinates requires

a dedicated kinematics from the user

�Advantage: freedom

�Drawback: laborious, risk of mistakes

� It is not limited to simplistic systems (with a good

library)

� No constraints in the equations of motion

� It is well adapted to education

Approach based on Minimal coordinates · acceleration (rotational and translational) of each body...

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