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Time-Optimal Path Tracking for Robots

a Convex Optimization Approach

Diederik Verscheure, Dr. B. Demeulenaere,Prof. J. Swevers, Prof. J. De Schutter, Prof. M. Diehl

Katholieke Universiteit Leuven,Department of Mechanical Engineering,

PMA Division

16 maart 2008

Outline

1 IntroductionTime-optimal motion planning

2 Time-optimal path trackingProblem formulationExisting solution methods

3 Reformulation and solution methodConvex reformulationDirect transcriptionSOCP formulationImplementation

4 Examples

5 Conclusions

Time-Optimal Path Tracking for Robots 2

Outline




4 Examples

5 Conclusions


Time-optimal motion planning


important for maximizing the productivityof robot systems.

reduces need for time-consuming manualprogramming.

Two approaches

direct approach.

decoupled approach.



The direct approach

solve motion planning directly in system’sstate space (2× n states).

complex because of nonlinear robotdynamics.

The decoupled approach

solve in two stages:

path planning: geometric.path tracking: dynamic (2 states).

easier and computationally cheaper.


Outline




4 Examples

5 Conclusions


Time-optimal path tracking


equations of motion

τ = M(q)q + C(q, q)q + Fs(q)sgn(q) + G(q),

path q(s), function of path coordinate s.

lower and upper bounds τ and τ ontorques.

Problem

Find relation s(t) that minimizes trajectoryduration T and satisfies torque constraints.



Path q(s) function of the path coordinate s.

q(s) = q′(s)s, (1)

q(s) = q′(s)s + q′′(s)s2, (2)

Equations of motion can be rewritten as

τ (s) = m(s)s + c(s)s2 + g(s), (3)

where

m(s) = M(q(s))q′(s), (4)

c(s) = M(q(s))q′′(s) + C(q(s),q′(s))q′(s), (5)

g(s) = Fs(q(s))sgn(q′(s)) + G(q(s)), (6)



Time-optimal path tracking subject to actuator constraints

minT ,s(·),τ (·)

T , (7)

subject to τ (t) = m(s(t))s(t) + c(s(t))s(t)2 + g(s(t)),

s(0) = 0 and s(T ) = 1, (8)

s(0) = s0 and s(T ) = sT , (9)

s(t) ≥ 0, (10)

τ (s(t)) ≤ τ (t) ≤ τ (s(t)), (11)

for t ∈ [0,T ].

Optimal control problem with 2 states s and s instead of 2× n.



Structure of the solution

velocity along the path s as high as possible.

at least one actuator saturated at each point on the path .

fully determined by “switching points” along the path.

0 2 4 6 8 100

2

4

time (s)

velo

city (

m/s

)

Velocity

0 2 4 6 8 10

−1

−0.5

0

0.5

1

time (s)

torq

ue (

N.m

)

Torque


Existing solution methods

Existing solution methods:

Indirect methods

Numerical searches and forward and backward integration todetermine “switching points”

tedious implementation.

exhaustive searches for candidate switching points.

not very flexible.

Other methods

Dynamic programming.

Direct methods.


Outline




4 Examples

5 Conclusions


Reformulation

Nonlinear optimal control problem

⇓

Convex optimal control problem

⇓

Discrete large scale convex optimizationproblem

⇓

Discrete large scale SOCP


Convex reformulation

objective function can be rewritten as

T =

∫ T

01dt =

∫ s(T )

s(0)

1

sds =

∫ 1

0

1

sds. (12)

introduce optimization variables

a(s) = s, (13)

b(s) = s2 (14)

and additional linear constraint

b′(s) = 2a(s). (15)




mina(·),b(·),τ (·)

∫ 1

0

1√b(s)

ds, (16)

subject to τ (s) = m(s)a(s) + c(s)b(s) + g(s), (17)

b(0) = s20 and b(1) = s2

T , (18)

b′(s) = 2a(s), (19)

b(s) ≥ 0, (20)

τ (s) ≤ τ (s) ≤ τ (s), (21)

for s ∈ [0, 1].

Convex objective function, linear constraints and time iseliminated.


Reformulation


⇓


⇓


⇓



Direct transcription

Direct transcription:

discretize the path coordinate s on [0, 1].

discretize b(s), a(s) and τi (s).

b(s) = s2 is assumed to be piecewise linear.

... s sK

sK−1

sK−2

s5

s4

s3

s2

s1

s0

bK

bK−1

bK−2

b5

b4

b3

b2

b1

b0

b(s)



objective function can be calculated analytically as∫ 1

0

1√b(s)

ds =K−1∑k=0

2∆sk

√bk+1 +

√bk

. (22)

differential constraint b′(s) = 2a(s) is rewritten as

(bk+1 − bk) = 2ak∆sk . (23)

. . .



Large-scale optimization problem

minak ,bk ,τ k

K−1∑k=0

2∆sk

√bk+1 +

√bk

,

subject to τ k = m(sk+1/2)ak + c(sk+1/2)bk+1/2 + g(sk+1/2),

b0 = s20 and bK = s2

T , (24)

(bk+1 − bk) = 2ak∆sk , (25)

bk ≥ 0 and bK ≥ 0, (26)

τ (sk+1/2) ≤ τ k ≤ τ (sk+1/2), (27)

for k = 0 . . .K − 1.


Reformulation


⇓


⇓


⇓



SOCP formulation

Second-order cone program

Standard form:

minx

fTx, (28)

subject to Fx = g, (29)

‖Mjx + nj‖2 ≤ pTj x + qj , (30)

for j = 1 . . .m.

Linear objective and equality constraints, and second-order coneinequality constraints.


SOCP formulation

SOCP formulation:

introduce variables dk .

new linear objective function

K−1∑k=0

2∆skdk . (31)

additional inequality constraints

1√bk+1 +

√bk

≤ dk . (32)


SOCP formulationSOCP formulation:

1√bk+1+

√bk≤ dk can be replaced by two equivalent

constraints by introducing variables ck

1

ck+1 + ck≤ dk (33)

ck ≤√

bk . (34)

can be rewritten as second order cone constraints∥∥∥∥ 2ck+1 + ck − dk

∥∥∥∥2

≤ ck+1 + ck + dk , (35)∥∥∥∥ 2ck

bk − 1

∥∥∥∥2

≤ bk + 1. (36)


SOCP formulation

SOCP formulation

minak ,bk ,τ k ,ck ,dk

K−1∑k=0

2∆skdk , (37)

subject to τ k = m(sk+1/2)ak + c(sk+1/2)bk+1/2 + g(sk+1/2),

b0 = s20 and bK = s2

T , (38)

(bk+1 − bk) = 2ak∆sk , (39)

τ (sk+1/2) ≤ τ k ≤ τ (sk+1/2), (40)∥∥∥∥ 2ck+1 + ck − dk

∥∥∥∥2

≤ ck+1 + ck + dk , (41)∥∥∥∥ 2ck

bk − 1

∥∥∥∥2

≤ bk + 1. (42)


SOCP formulation

OK, but...

this sounds all very complicated...

and why should I care?

Easy and efficient

Once the (complicated) formulation is done, implementation is

very easy.

automatically efficient.

requires almost no optimization knowledge.


Matlab implementation

Matlab implementation using YALMIP:

objective function∑K−1

k=0 2∆skdk

t = sum(2*dvar.*ds);

constraints ∥∥∥∥ 2ck

bk − 1

∥∥∥∥2

≤ bk + 1 (43)

F = F + set(cone([2*cvar(1,k); bvar(1,k)-1], bvar(1,k)+1));

solve with SeDuMidiagnostics = solvesdp(F,t);

no numerical “hassle”: tolerances, scaling, . . .

implementation straightforward, solution automaticallyefficient.


Outline




4 Examples

5 Conclusions


Three-DOF elbow manipulator

Example:



Torques:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−150

−100

−50

0

50

100

150

path coordinate (−)

torq

ue (

N.m

)

joint torque 1joint torque 2joint torque 3



Numerical jitter

solution (acceleration) is not uniquely defined (singular point).

major problem for indirect methods. Singular arcs can resultin infinitely many “switching points”.

not a problem for the solver.

Undesirable in practice

just add a suitable penalty term to the objective function∫ T

0|τi (s)| dt =

∫ 1

0

|τ ′i (s)s|s

ds =

∫ 1

0

∣∣τ ′i (s)∣∣ ds, (44)

still SOCP.



Torques with penalization of torque jumps:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−150

−100

−50

0

50

100

150

path coordinate (−)

torq

ue (

N.m

)

joint torque 1joint torque 2joint torque 3


Complicated example

A more complicated example: a six-DOF manipulator carrying outan non-smooth path. Discussed on poster.

q1

q2

q3

X

Y

Z


Outline




4 Examples

5 Conclusions


Conclusions

Conclusions:

convex optimization approach to time-optimal path tracking.

easy to implement.

very efficient.

flexible.

Future work:

use for real-time motion generation.

integration with path planning.

exploration of various applications, such as minimization ofreaction forces at the robot base.


Thank you for your attention

Thank you for your attention!

Comments/Suggestions/Questions?
