Robotics 2

Prof. Alessandro De Luca

Dynamic model of robots: Newton-Euler approach

Approaches to dynamic modeling (reprise)

energy-based approach (Euler-Lagrange)

!  multi-body robot seen as a whole

!  constraint (internal) reaction forces between the links are automatically eliminated: in fact, they do not perform work

!  closed-form (symbolic) equations are directly obtained

!  best suited for study of dynamic properties and analysis of control schemes

Newton-Euler method (balance of forces/torques)

!  dynamic equations written separately for each link/body

!  inverse dynamics in real time

!  equations are evaluated in a numeric and recursive way

!  best for synthesis (=implementation) of model-based control schemes

!  by elimination of reaction forces and back-substitution of expressions, we still get closed-form dynamic equations (identical to those of Euler-Lagrange!)

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Derivative of a vector in a moving frame

!i

derivative of “unit” vector

… from velocity to acceleration

!

0 ˙ R i = S 0" i( )0Ri

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Dynamics of a rigid body

!

fi" =ddt

mvc( ) = m ˙ v c

!  Newton dynamic equation !  balance: sum of forces = variation of linear momentum

!  Euler dynamic equation !  balance: sum of torques = variation of angular momentum

!  principle of action and reaction !  forces/torques: applied by body i to body i+1 = - applied by body i+1 to body i

!

µi" =ddt

I#( ) = I ˙ # +ddt

R I R T( )# = I ˙ # + ˙ R I R T + R I ̇ R T( )#= I ˙ # + S #( )R I R T# + R I R TST #( )# = I ˙ # +# $ I#

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Newton-Euler equations - 1

link i

axis i (qi)

vci

axis i+1 (qi+1)

Oi-1 Oi . .

fi fi+1

ci zi

mig

fi force applied from link (i-1) on link i

fi+1 force applied from link i on link (i+1)

Newton equation

mig gravity force

all vectors expressed in the same RF (better RFi)

FORCES

N

zi-1

linear acceleration of ci

center of mass

!

fi" fi+1 + mig = miaci

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Newton-Euler equations - 2 link i !i

Oi-1 Oi . .

"i "i+1 ri-1,ci ri,ci

"i torque applied from link (i-1) on link i

"i+1 torque applied from link i on link (i+1)

Euler equation

TORQUES

fi fi+1

fi x ri-1,ci torque due to fi w.r.t. ci

- fi+1 x ri,ci torque due to - fi+1 w.r.t. ci

E

zi zi-1

all vectors expressed in the same RF (RFi !!)

Robotics 2 6 angular acceleration of bodyi

axis i (qi)

axis i+1 (qi+1)

gravity force gives no torque at ci

Forward recursion Computing velocities and accelerations

•  “moving frames” algorithm (as for velocities in Lagrange) •  wherever there is no leading superscript, it is the same as the subscript •  for simplicity, only revolute joints (see textbook for the more general treatment) initializations

AR

the gravity force term can be skipped in Newton equation, if added here Robotics 2 7

Backward recursion Computing forces and torques

at each step of this recursion, we have two vector equations (Ni + Ei) at the joint providing and : these contain ALSO the reaction forces/torques at the joint axis ⇒ they should be next “projected” along/around this axis

from Ni to Ni-1

from Ei to Ei-1

F/TR

initializations

add here dissipative terms (here viscous friction only)

eliminated, if inserted in forward recursion (i=0)

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generalized forces (in rhs of Euler-Lagrange eqs)

for prismatic joint

for revolute joint N scalar

equations FP

at the end

ri,ci ri,ci)

Comments on Newton-Euler method

!  the previous forward/backward recursive formulas can be evaluated in symbolic or numeric form !  symbolic

!  substituting expressions in a recursive way !  at the end, a closed-form dynamic model is obtained, which

is identical to the one obtained using Euler-Lagrange (or any other) method

!  there is no special convenience in using N-E in this way !  numeric

!  substituting numeric values (numbers!) at each step !  computational complexity of each step remains constant ⇒

grows in a linear fashion with the number N of joints (O(N)) !  strongly recommended for real-time use, especially when the

number N of joints is large

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Newton-Euler algorithm efficient computational scheme for inverse dynamics

AR

AR

F/TR

F/TR

FP

FP

inputs outputs

(force/torque exchange environment/E-E)

(at robot base) numeric steps at every instant t

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!  data file (of a specific robot) !  number N and types ! ={0,1}N of joints (revolute/prismatic) !  table of DH kinematic parameters !  list of dynamic parameters of the links (and of the motors)

!  input !  vector parameter " = {0g,0} (presence or absence of gravity) !  three ordered vector arguments

!  typically, samples of joint position, velocity, acceleration taken from a desired trajectory

!  output !  generalized force u for the complete inverse dynamics !  … or single terms of the dynamic model

general routine NE"(arg1,arg2,arg3)

Matlab (or C) script

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Examples of output

!  complete inverse dynamics

!  gravity terms

!  centrifugal and Coriolis terms

!  i-th column of the inertia matrix

!  generalized momentum

ei = i-th column of identity matrix

u = NE0g(q,0,0) = g(q)

u = NE0(q,0,ei) = bi(q)

u = NE0(q,0,q) = B(q)q . .

u = NE0(q,q,0) = c(q,q) . .

u = NE0g(qd,qd,qd) = B(qd)qd+c(qd,qd)+g(qd)=ud

. . .. ..

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Inverse dynamics of a 2R planar robot

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desired (smooth) joint motion: quintic polynomials for q1, q2 with zero vel/acc boundary conditions

from (90°,-180°) to (0°,90°) in T=1 s

⇔

Inverse dynamics of a 2R planar robot

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motion in vertical plane (under gravity) both links are thin rods of uniform mass m1 = 10 kg, m2 = 5 kg

zero initial torques = free equilibrium configuration

+ zero initial

accelerations

final torques u1 ! 0, u2 = 0

balance link weights

in final (0°,90°) configuration

Inverse dynamics of a 2R planar robot

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torque contributions at the two joints for the desired motion = total, = inertial = Coriolis/centrifugal, = gravitational

Use of NE routine for simulation direct dynamics

!  numerical integration, at current state (q,q), of

!  Coriolis, centrifugal, and gravity terms

!  i-th column of the inertia matrix, for i =1,..,N

!  numerical inversion of inertia matrix

!  given u, integrate acceleration computed as

bi = NE0(q,0,ei)

InvB = inv(B)

q = B-1(q) [u – (c(q,q)+g(q))]= B-1(q) [u – n(q,q)] . ..

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.

complexity O(N)

O(N2)

O(N3) but with small coefficient

n = NE0g(q,q,0)

.

.

q = InvB * [u – n] .. new state (q,q)

and repeat over time...

.