On the Dynamics and Optimal Control of Constrained Mechanical Systems

On the Dynamics and Optimal Control of Constrained Mechanical Systems

Simeon Schneider, Peter Betsch Institute of Mechanics

Karlsruhe Institute of Technology (KIT) Otto-Ammann-Platz 9, 76131 Karlsruhe, Germany

simeon.schneider@kit.edu, peter.betsch@kit.edu ABSTRACT

The focus of this work is on optimal control in redundant coordinates with a special attention to the boundary constraints that arise in this context. Due to the similarity of the optimization problem of optimal control to the Lagrangian formalism of clas- sical mechanics, this is considered first. Once the mechanical problem of the bound- ary conditions in redundant coordinates has been discussed, the knowledge gained is transferred to the optimal control problem in order to solve the problem in redun- dant coordinates. Finally, for each section the equivalence of the problem in minimal coordinates and redundant coordinates is shown by numerical results.

Keywords:Optimal control, boundary value problems, multibody dynamics 1 INTRODUCTION

Optimal control contains a large field of applications, from the optimal control of economical or chemical processes to the optimal control of robots or satellites. The latter ones can be summa- rized to mechanical systems and represent the focus of attention within this work.

There are several methods to describe the behaviour of the systems, the two most frequently cho- sen being the description of the complete system in minimal or redundant coordinates. However, depending on the choice of the description of the system, different types of mathematical systems of equations arise. As is well known, the description of the system in minimal coordinates yields a system of ordinary differential equations (ODEs), whereas the description of the system in redun- dant coordinates yields a system of differential algebraic equations (DAEs) for the equations of motion. Of course, both systems of equations describe the same motion of the mechanical system.

Since the equations of motion are the constraints in the optimal control problem of the mechanical system, their description plays an essential role in the formulation of the optimal control problem.

While the analytical and numerical solution of the optimal control problem in minimal coordinates can be considered as already well researched, this is not yet true for the description in redundant coordinates. However, since finding minimal coordinates can be difficult and in the worst case impossible, the focus of the present work lies on using DAEs as state equations in the optimal control problem. Due to the parallels between the Lagrangian formalism of classical mechanics and the optimization problem of optimal control, the Lagrangian formalism will be considered first. Subsequently, the obtained knowledge is applied to the optimal control problem, with special attention to the boundary constraints to be defined.

To keep it short and simple, the general approach will be illustrated with the example of a physical pendulum on a slide depicted in Figure 1.

2 CONSTRAINED MECHANICS AND THE LAGRANGIAN FORMALISM

In this section the boundary value problem (BVP) of the Lagrangian formalism will be consid- ered with a special attention to the boundary conditions arising in redundant coordinates. Before starting with the description of the BVP in redundant coordinates, however, we will briefly discuss

the BVP in minimal coordinates. Once the procedure is known, the BVP is derived in redun- dant coordinates and the problems which arise are discussed. Thereupon, a simple but promising way to solve the mechanical BVP in redundant coordinates is described. The equivalence of both boundary value problems under consideration is illustrated with a numerical example.

2.1 The mechanical BVP in minimal coordinates

Let I ∈[t0,tf] be a time interval, q:I →M be the minimal coordinates on the configuration manifold M :={q∈R²|q1 =x1,q2=x6} of the system at hand (Fig. 1), and T M be the tangent bundle, see for example [2]. Then the LagrangianL:T M→Ris defined by

L(q) =T(q,q)˙ −V(q) =1

2q˙^agabq˙^b−V(q)

withT:T M →Rbeing the kinetic energy,V :M→Rbeing the potential energy andgabbeing the metric tensor, better known as mass matrix. Finally letP∈M×T^∗M be the phase space of the mechanical system. Then using Hamilton’s principle, the mechanical BVP is described by the optimization problem

Problem 1. Mechanical BVP in minimal coordinates (MBVPM) Find the extremal curveγ:I→Pgiven by the action integral

S^M(q) = Z _tf

t0

L(q,q˙)dt (1)

Using Livens principle, see [3], [4], one may also set the equivalent problem to (1) by Problem 2. Extended mechanical BVP in minimal coordinates (MBVPML) Find the extremal curveγ:I→Pgiven by the action integral

S^M(q,v,p) =^{Z tf}

t₀ L(q,v)−pa(v^a−q˙^a)dt (2) withv∈T M and andp∈T^∗M.

Taking the variation of (2) yields δS^M(q,v,p) =^{Z tf}

t₀ (∂L(q,v)

∂v^a −pa)δv^a+ (∂L(q,v)

∂q^a )δq^a+ (q˙^a−v^a)δpa+paδq˙^adt (3) Applying the integration by parts

Z _tf

t₀ paδq˙^adt= [paδq^a]^t_t0^f − Z _tf

t₀ p˙aδq^adt (4)

x1

x2

x3

x4

x5

x6

m1,2l1

m2,2l2

Figure 1. Multibody system of a physical pendulum on a moving slide taken from [1].

and inserting the Legendre transformation resulting from the variationδv^a, which yield the Leg- endre transformed functionH^M:T^∗M→Rof the Lagrangian,

H^M=1

2pag^abpb+V(q) (5)

finally yields

δS^M(q,p) =^{Z tf}

t0

(−p˙a−∂H^M(q,p)

∂q^a )δq^a+ (q˙^a−∂H^M(q,p)

∂pa )δpadt+ [paδq^a]^t_t0^f (6) Now, imposing the stationary conditionδS^M(q,p) =0, one obtains the necessary optimality con- ditions

˙

q^a=∂H^M(q,p)

∂pa

˙

pa=−∂H^M(q,p)

∂q^a

(7)

along with the boundary conditions

0= [paδq^a]^t_t0^f (8)

Note that, at this stage, we do not impose the common end-point conditions onδq^a, but rather keep the boundary conditions leading to a BVP comprised of (7) and (8).

2.2 The mechanical BVP in redundant coordinates

Let x:I →M be the description of the mechanical system in redundant coordinates and thus M :={x∈R⁶|g^r(x) =0}. Then the Lagrangian in redundant coordinates may be defined by

L(x,ˆ y) =T(x,x)˙ −V(x)−yrg^r(x) =1

2x˙ⁱgˆiax˙^j−V(x)−yrg^r(x) (9) In analogy to the last section we obtain

Problem 3. Extended mechanical BVP in redundant coordinates (MBVPRL) Find the extremal curveγ:I→Pgiven by the action integral

S^M(x,v,ˆ p,y) =ˆ ^{Z tf}

t0

L(x,ˆ v,ˆ y)−pˆi(vˆⁱ−x˙ⁱ)dt (10) Following the steps which led to (3) - (6), we obtain

δS^M(x,pˆ,y) = Z _tf

t0

(∂Hˆ(x,pˆ,y)

∂xⁱ −p˙ˆi)δxⁱ+ (x˙ⁱ−∂Hˆ(x,pˆ,y)

∂pˆi )δpˆi+g^r(x)δyrdt+ ˆ piδxⁱtf

t₀ (11) where the Hamiltonian ˆH^Min redundant coordinates is given by

Hˆ^M(x,pˆ,y) =pˆigˆ^{i j}pˆj+V(x) +yrg^r(x) (12) Thus, the mechanical BVP in redundant coordinates is described by the necessary optimality con- ditions

˙

xⁱ=∂Hˆ^M(x,p,ˆ y)

∂pi

˙ˆ

pi=−∂Hˆ^M(x,p,y)ˆ

∂xⁱ 0=∂Hˆ^M(x,p,ˆ y)

∂yr

(13)

along with the boundary conditions

0= ˆ piδxⁱt_f

t0 (14)

However, due to the constraintsg^r(x) =0 associated with (13)3, the variations δxare not inde- pendent and shall be further investigated with respect to the boundary conditions. Since (13) gives rise to index-3 DAEs, see [5], it is well known, that the implicit definition ofyris given by

d²

dt²(g^r(x)) =G^r(x,p,y) (15) However, since eitherxⁱ or pi is properly defined on the boundary, yr is in general unknown on the boundaries, until the solution is known. Thusg^r(x(t)) =0 holds on the boundaries, which reduces the independent redundant coordinates toaand thus onlyaboundary conditions may be defined with respect to the variation ofxon the boundaries. To answer the question of properly set admissible boundaries, it is helpful to decompose the redundant coordinates into their normal and tangential parts. This is to happen in the following section.

2.2.1 Decomposed vector spaces

LetBVˆ_p andBVˆ_p^∗ be the bases of the local vector spaces ˆVpand ˆV_p^∗at pointponM in redundant coordinates. Let furtherBTpM andBT_p^∗M be the bases of the tangential spacesTpMandT_p^∗M, andBNpM andBN_p^∗M be the bases of the normal spaces NpM and N_p^∗M at p. Now let’s assume, the mappingn^r(x) =const.exists and thus g^r(n^s(x)) =0 holds. Since the constraints g^r(x) are known in redundant coordinates, following [1] , see also [6], [7] and [8], a straight forward calculation yields the metric tensors associated with the normal and tangential spaces, (gˆrs,gˆ^rs), and(gab,g^ab), respectively, along with the Jacobians

∂xⁱ

∂n^r : ˆVp→NpM, ∂n^r

∂xⁱ : ˆV_p^∗→N ∗pM, (16)

∂xⁱ

∂q^a : ˆVp→TpM, ∂q^a

∂xⁱ : ˆV_p^∗→T ∗pM, (17) Now let the bases of the decomposed local vector spaceVp:=NpM∪TpM and its dual space V_p^∗:=N ∗pM∪T ∗pM be defined byBV_p andBV_p^∗, whereat by definition

BT_pM ⊥BN_pM, BT_p^∗M ⊥BN_p^∗M (18) holds. This leads to the mappings

∂x^j

∂zⁱ : ˆVp→NpM∪TpM (19)

∂z^j

∂xⁱ : ˆV_p^∗→N ∗pM∪T ∗pM (20) where the coordinateszⁱ={n^r,q^a}and momentapi={pr,pa}have been introduced.

Remark. Velocity components in normal directions Since

dg^r(x)

dt =∂g^r(x)

∂n^s n˙^s=0 (21)

and^∂g_∂^r_n^(x)s 6=0in general, it can be seen thatn˙^s=0holds.

2.2.2 The boundary conditions in redundant coordinates Using the relationships

δxⁱ= ∂xⁱ

∂z^jδz^j= ∂xⁱ

∂n^rδn^r+∂xⁱ

∂q^aδq^a (22)

ˆ pi=∂z^j

∂xⁱpj=∂n^r

∂xⁱpr+∂q^a

∂xⁱ pa (23)

inserting them into (11) and making use of ˙n^r=pr=0 leads to the necessary optimality conditions in decomposed coordinates given by

˙

q^a=∂H^M(q,n,p)

∂pa (24a)

0=−1 2pa∂g^ab

∂n^r pb−∂V

∂n^r−yr (24b)

˙

pa=−∂H^M(q,n,p)

∂q^a (24c)

0=g^r(x) (24d)

with the Hamiltonian being defined by (5). For the boundary conditions follow 0=

pi∂xⁱ

∂n^rδn^r t_f

t0

+

pi∂xⁱ

∂q^aδq^a t_f

t0

= [prδn^r]^t_t0^f + [paδq^a]^t_t0^f

= [paδq^a]^t_t0^f

(25)

sincepr=0 everywhere.

Remark. The implicit definitions ofyr

It can be seen from(24b)that r ODEs in redundant coordinates correspond to the implicit defini- tion of yr.

Remark. The duality of the derivative of the constraints and the variation forδn^r A straight forward calculation yields

d²

dt²(g^r(x)) =G^r(x,p,y) =ˆ g^rsGs(q,n,p,y) (26) with

Gr(q,n,p,y) =−1 2pa∂g^ab

∂n^r pb−∂V

∂n^r−yr (27)

Writing (25) in terms of redundant coordinates finally yields the proper BVP in redundant coordi- nates, defined by the necessary optimality conditions (13) and the boundary conditions

0=

ˆ pi∂xⁱ

∂q^aδq^a t_f

t₀ (28)

Remark. Setting boundary conditions by using Lagrangian multipliers

The natural boundary conditions(28)may also be augmented with suitable end-point conditions by using Lagrangian multipliers in the action integral. For example, this leads to

S^M(·) = Z t_f

t₀ L(q,q)˙ dt+µa⁰(q^a(t0)−q¯^a₀) +µa^N(q^a(tf)−q¯^a_N) (29) and

S^M(·) = Z _tf

t0

L(x,x,y)˙ dt+µ_a⁰∂q^a

∂xⁱ(xⁱ(t0)−x¯ⁱ₀) +µ_a^N∂q^a

∂xⁱ(xⁱ(tf)−x¯ⁱ_N) (30) whereq¯^a₀,q¯^a_Nandx¯ⁱ₀,x¯_Nⁱ , respectively, are prescribed coordinates.

2.3 Numerical example of the mechanical BVP

In the following, the BVP of the mechanical system depicted in Fig. 1 is solved in minimal coordinates and redundant coordinates to show the equivalence of the BVPs numerically. As can be observed from Fig. 1, the constraints are specified by

g¹(x) =x² (31a)

g²(x) =x³ (31b)

g³(x) =x¹−x⁴+l2sin(x⁶) (31c) g⁴(x) =x²−x⁵+l2cos(x⁶) (31d) Let the potential energy in redundant and minimal coordinates be defined by

V(x) =−m1gx²−m2gx⁵, V(q) =−m2gl2cos(q²) (32) withgbeing the gravitational constant. Let further the kinetic energy be given by

T(x,p) =1

2pⁱgˆi jp^j, T(q,n,p) =1

2p^agabp^b (33)

with the metric tensors

ˆ

gi j=Mˆ =











m1 0 0 0 0 0

0 m1 0 0 0 0

0 0 ^m1(2l₁₂¹⁾² 0 0 0

0 0 0 m2 0 0

0 0 0 0 m2 0

0 0 0 0 0 ^m2(2l₁₂²⁾²











(34)

and

gab=M=

m1+m2 m2l2cos(q²) m2l2cos(q²) ⁴₃m2l₂²

(35) Applying the midpoint rule to the differential part, with step sizeh=tn+1−tn, finally yields the discrete necessary optimality conditions in redundant coordinates as

xⁱ_n+1−xⁱ_n = h ^∂HM

∂pˆⁿ⁺_i ¹²; n = {0,1, ...,N−1}

ˆ

pⁿ⁺¹_i −pˆⁿ_i = −h_∂^∂_x^Hi ^M n+1 2

; n = {0,1, ...,N−1}

0 = g^r(xn); n = {0,1, ...,N}

(36)

along with the boundary conditions

∂q^a₀

∂xⁱ₀(xⁱ₀−x¯ⁱ₀)δ µa⁰ = 0; ^∂_∂^q_x^aNi

N(xⁱ_N−x¯ⁱ_N)δ µa^N = 0 (^∂_∂^x_qⁱ⁰a

0pˆ⁰_i −µa⁰)δq^a₀ = 0; (^∂_∂q^xiNa

Npˆ^N_i −µa^N)δq^a_N = 0 (37)

Since the implicit definition ofyrcan’t be done properly on the boundary by the discrete system of ODEs, the implicit definitions are enforced by making use of (15). Thus, the additional boundary conditions

G^r(x0,pˆ0,y0) =0; G^r(xN,pˆN,yN) =0 (38)

¯

x(t0) = [0,0,0,1,0, ^π₂]^T

¯

x(tf) = [0,0,0,−1,0,−^π₂]^T (m1,m2) = (1,1)

(l1,l2) = (1,1) T = 1.8

N = 100

Table 1. Specified boundary conditions as well as physical and geometric parameters of the mechanical system.

0 0.5 1 1.5 2

time t -2

-1 0 1 2

xi

MIN RED

0 0.5 1 1.5 2

time t -4

-2 0 2 4

pi

MIN RED

0 0.5 1 1.5 2

time t -60

-40 -20 0 20 40

yi

MIN RED

Figure 2. Solutions of the mechanical boundary value problem defined by Table 1. Here the positions (left) and momenta (center) are shown in redundant coordinates. The dual quantities yrin minimal coordinates are calculated by making use of (24b)

are enforced which ensure the definition ofyron the boundaries.

For the discrete necessary optimality conditions in minimal coordinates follows q^a_n+1−q^a_n = h ^∂HM

∂pⁿ⁺a ¹²

; n ={0,1, ...,N−1}

pⁿ⁺¹_a −pⁿ_a = −h_∂^∂_q^Ha^M n+1 2

; n ={0,1, ...,N−1} (39)

along with the boundary conditions

(q^a₀−q¯^a₀)δ µa⁰ = 0; (q^a_N−q¯^a_N)δ µa^N = 0

(p⁰_a−µ_a⁰)δq^a₀ = 0; (p^N_a −µ_a^N)δq^a_N = 0 (40) The boundary conditions chosen for the example, the length and mass of the slide and the pendu- lum as well as the length of the time intervalI ∈[0,T]and the number of discrete time intervals, N, is shown in Table 1

Comparing the solutions depicted in Fig. 2, it can be seen that the BVPs are indeed equivalent.

Fig. 3 also shows a sequence of the solution of the mechanical boundary value problem.

3 OPTIMAL CONTROL OF CONSTRAINED MECHANICAL SYSTEMS USING RE- DUNDANT COORDINATES

In this section the BVP arising from the optimal control problem shall be treated by using redun- dant coordinates. Therefore, the well known optimal control problem in minimal coordinates will be discussed briefly to show the close connection between the Lagrange formalism in the mechan- ical BVP and the optimal control problem. Afterwards the necessary optimality conditions for the optimal control problem in redundant coordinates proofed in [9] shall be viewed. The boundary

-2 0 2 x-axis -3

-2 -1 0 1 2 3

y-axis

t = 0

-2 0 2

x-axis -3

-2 -1 0 1 2 3

y-axis

t = 0.378

-2 0 2

x-axis -3

-2 -1 0 1 2 3

y-axis

t = 0.756

-2 0 2

x-axis -3

-2 -1 0 1 2 3

y-axis

t = 1.134

-2 0 2

x-axis -3

-2 -1 0 1 2 3

y-axis

t = 1.512

-2 0 2

x-axis -3

-2 -1 0 1 2 3

y-axis

t = 1.8

Figure 3. Snapshots of the motion resulting from the BVP with data given in Table 1.

conditions for the redundant coordinates will be described and finally the equivalence between the BVPs in terms of redundant and minimal coordinates will be shown with a numerical example.

3.1 The optimal control BVP in minimal coordinates

Let the controlled equations of motion be defined by (7) together with the control forcesu:I→R² and thus

˙

qⁱ=∂H^M

∂pa , p˙a=−∂H^M

∂q^a +ua (41)

Then the standard optimal control problem reads

Problem 4. Optimal control problem in minimal coordinates (OCBVPM) Minimize

S^OC(q,p,u) =^{Z tf}

t0

C(q,p,u)dt (42)

subject to

˙

q^a= ∂H^M

∂pa

˙

pa=−∂H^M

∂q^a +ua

(43)

Augmenting the objective function of the optimization problem with the dynamic constraints (43) yields

Problem 5. Augmented optimization problem in minimal coordinates (OCBVPML) Extremize

S^OC(q,p,u,λ^q,λp) = Z _tf

t₀ C(q,p,u) +λa^q

˙

q^a−∂H^M

∂pa

+λp^a

˙

pa−(−∂H^M

∂q^a +ua)

dt (44)

= Z _tf

t₀ λa^qq˙^a+λp^ap˙a−H^OC(·)dt (45)

Here, the Hamiltonian of the optimal control problem H^OC(q,p,u,λ^q,λp) =λa^q∂H^M

∂pa +λp^a(−∂H^M

∂q^a +ua)−C(q,p,u) (46)

hast been introduced, see e.g. [8], [10] , [11]. Following the Lagrangian formalism by taking the variation and then applying integration by parts for the termsλ_i^qδx˙ⁱandλ_p^aδp˙afinally yields

δS^OC(q,p,u,λ^q,λp) = Z _tf

t0

δλ_a^q

˙

q^a−∂H^OC(·)

∂λa^q

+δλ_p^a p˙a−∂H^OC(·)

∂λ_p^a

!

+δq^a(−λ˙_a^q−∂H^OC(·)

∂q^a ) +δpa(−λ˙_p^a−∂H^OC(·)

∂pa ) +δua∂H^OC(·)

∂ua dt + [λ_a^qδq^a]^t_t0^f +

λ_p^aδpat_f t0

(47) Thus, the necessary optimality conditions using minimal coordinates yield

˙

q^a=∂H^OC(·)

∂λa^q , λ˙_a^q=−∂H^OC(·)

∂q^a

˙

pa=∂H^OC(·)

∂λ_p^a , λ˙p^a=−∂H^OC(·)

∂pa

0=∂H^OC(·)

∂ua

(48)

along with the boundary conditions

0= [λ_a^qδq^a]^t_t0^f, 0=

λ_p^aδpatf

t₀ (49)

3.2 The necessary optimality conditions in redundant coordinates

Using redundant coordinates for the controlled equations of motion, the optimal control problem under investigation is given by

Problem 6. Optimal control problem in redundant coordinates (OCBVPR) Minimize

Sˆ^OC(x,p,y,ˆ u) =ˆ ^{Z tf}

t₀ C(x,p,ˆ y,u)ˆ dt (50)

subject to

˙

xⁱ= ∂Hˆ^M

∂pˆi

p˙ˆi=−∂Hˆ^M

∂xⁱ +uˆi

0=g^r(x)

(51)

with ˆu:I→R⁶. Again augmenting the objective function with the dynamic constraints (51) yields S¯^OC(x,p,ˆ y,u,ˆ λˆ^q,λˆp,η) =^{Z tf}

t₀ C(x,p,y,ˆ u) +λˆ_i^q(x˙ⁱ−∂Hˆ^M

∂pˆi ) +λˆ_pⁱ(p˙ˆi−(−∂Hˆ^M

∂xⁱ +uˆi)) +ηrg^r(x)dt

= Z t_f

t₀

λˆ_i^qq˙ⁱ+λˆqⁱp˙ˆi−H¯^OC(x,p,y,ˆ u,ˆ λˆ^q,λˆp,η)dt (52) with

H¯^OC(x,p,y,ˆ u,ˆ λˆ^q,λˆp,η) =λˆ_i^q∂Hˆ^M

∂pˆi +λˆ_pⁱ(−∂Hˆ^M

∂xⁱ +uˆi)−ηrg^r(x)−C(x,p,y,u)ˆ (53)

One might now again demand

δS¯^OC(x,p,y,ˆ u,ˆ λˆ^q,λˆp,η) =0

to get the necessary optimality conditions. However, it is proofed in [9] (see also [10]), that this approach is not feasible. Instead, following [9], the proper necessary optimality conditions for the optimal control problem in redundant coordinates are given by

˙

xⁱ=∂Hˆ^OC(·)

∂λˆ_i^q , λ˙ˆ_i^q=−∂Hˆ^OC(·)

∂xⁱ p˙ˆi=∂Hˆ^OC(·)

∂λˆ_pⁱ , λ˙ˆ_pⁱ =−∂Hˆ^OC(·)

∂pˆi

0=g^r(x), 0=∂Hˆ^OC(·)

∂yr

0=∂Hˆ^OC(·)

∂uˆi

(54)

with the Hamiltonian being

Hˆ^OC(x,p,y,ˆ u,ˆ λˆ^q,λˆp,η) =λˆ_i^q∂H^M

∂pˆi +λˆ_pⁱ(−∂H^M

∂xⁱ +uˆi)−ηrG^r(x,p,y,ˆ u)ˆ −C(x,p,y,ˆ u)ˆ (55) whereatG^r(x,pˆ,y,uˆ) corresponds to the implicit definition of the dual normal quantity yr. For index-3 DAEs one has

G^r(x,p,y,ˆ u) =ˆ d²g^r(x)

dt² (56)

3.3 The optimal control problem in redundant coordinates

Even though the proper necessary optimality conditions in redundant coordinates are given by [9]

(see also [10]), the treatment of the boundary conditions for (OCBVPR) still demands further elab- oration. Comparing (53) and (55), it can be seen that the natural boundary conditions nevertheless arise from integration by parts:

Z _tf

t0

λˆ_i^qδx˙ⁱdt=h

λˆ_i^qδxⁱitf

t₀− Z _tf

t0

λ˙ˆ_i^qδxⁱdt (57) Z _tf

t₀

λˆ_pⁱδp˙ˆidt=h λˆ_pⁱδpi

it_f t₀−

Z _tf

t₀

λ˙ˆ_pⁱδpˆidt (58) and thus the proper optimal control BVP is initially given by the necessary optimalty conditions (54) along with the boundary conditions

hλˆ_i^qδxⁱit_f t0

=0 h

λˆ_pⁱδpˆi

it_f t0

=0 (59)

However, sincepr=0 andn^r(x) =const. has to hold everywhere and especially on the boundaries, the corresponding variations have to vanish. Making use of (19) and (20), the boundary conditions in redundant coordinates reduce to

0=

λˆ_i^q(∂xⁱ

∂n^rδn^r+ ∂xⁱ

∂q^aδq^a) t_f

t₀

= [λr^qδn^r+λa^qδq^a]^t_t0^f = [λa^qδq^a]^t_t0^f (60) 0=

λˆ_pⁱ(∂n^r

∂xⁱδpr+∂q^a

∂xⁱδpa) t_f

t₀

=

λ_p^rδpr+λ_p^aδpatf

t₀ =

λ_p^aδpatf

t₀ (61)

Accordingly, the proposed form of the optimal control BVP in redundant coordinates is given by the necessary optimality conditions (54) together with the boundary conditions

0=

λˆ_i^q∂xⁱ

∂q^aδq^a tf

t 0=

λ_pⁱ∂q^a

∂xⁱδpa

tf

t (62)

3.4 Numerical example of the optimal control BVP Let the cost functionals be defined by

C(x,p,y,ˆ u) =ˆ 1

2uˆigˆ^{i j}uˆj, C(q,p,u) =1 2uag^{i j}ua

with the metric tensors given by ˆ

g^{i j}=Mˆ ⁻¹, g^ab=M⁻¹

Using once more the midpoint rule for the discretization of the differential equations in the neces- sary optimality conditions in term of redundant coordinates yields

xⁱ_n+1−xⁱ_n = h ^∂HˆOC

∂λ^ˆ_i^q,n+12; n ={0,1, ...,N−1} pⁿ⁺¹_i −pⁿ_i = h_∂^∂_λˆ^Hˆ_i^OC

p,n+1 2

; n ={0,1, ...,N−1} λˆ_i^q,n+1−λˆ_i^q,n = h(−_∂^∂_x^Hˆi^OC

n+1 2

); n ={0,1, ...,N−1}

λˆ_p,n+1ⁱ −λˆ_p,nⁱ = h(−^∂HˆOC

∂pˆⁿ⁺_i ¹²); n ={0,1, ...,N−1} 0 = g^r(xⁱ_n); n ={0,1, ...,N} 0 = ^∂Hˆ_∂^OC_yn^(·)

r ; n ={0,1, ...,N} 0 = ^∂Hˆ_∂^OC_uˆn^(·)

i ; n ={0,1, ...,N}

(63)

As already done in the context of the mechanical BVP, the boundary conditions may also be enforced by Lagranian multipliers. Consequently, we get

∂q^a₀

∂xⁱ₀(xⁱ₀−x¯ⁱ₀)δ µa⁰ = 0; ^∂_∂^q_xi^aN

N(x_Nⁱ −x¯ⁱ_N)δ µa^N = 0

∂xⁱ₀

∂q^a₀(p⁰_i −p¯⁰_i)δν₀^a = 0; ^∂_∂^x_q^iNa

N(p^N_i −p¯^N_i )δν_N^a = 0 (^∂_∂^x_qⁱ⁰a

0λ_i^q,0−µ_a⁰)δq^a₀ = 0; (^∂_∂q^xiNa

Nλ_i^q,N−µ_a^N)δq^a_N = 0 (^∂_∂^q_xi^a0

0λ_p,0ⁱ −ν₀^a)δp⁰_a = 0; (^∂_∂^q_x^aN_i

Nλ_p,Nⁱ −ν_N^a)δp^N_a = 0

(64)

together with the implicit definition of the dual quantitiesyron the boundaries already known from the mechanical BVP given by (56).

In minimal coordinates, the discrete necessary optimality condition simplify to q^a_n+1−q^a_n = h ^∂HOC

∂λa^q,n+12

; n = {0,1, ...,N−1}

pⁿ⁺¹_a −pⁿ_a = h_∂λ^∂Ha^OC p,n+1

2

; n = {0,1, ...,N−1}

λa^q,n+1−λa^q,n = h(−^∂_∂^H_q^{OCa (·)}

n+1 2

); n = {0,1, ...,N−1}

λ_p,n+1ⁱ −λ_p,nⁱ = h(−^∂HOC(·)

∂pⁿ⁺_i ¹² ); n = {0,1, ...,N−1}

0 = ^∂H_∂^OC_un^(·)

a ; n = {0,1, ...,N}

(65)

along with the boundary conditions

(q^a₀−q¯^a₀)δ µ_a⁰ = 0; (q^a_N−q¯^a_N)δ µ_a^N = 0;

(p⁰_a−p¯⁰_a)δν₀^a = 0; (p^N_a −p¯^N_a)δν_N^a = 0;

(λa^q,0−µ_a⁰)δq^a₀ = 0; (λa^q,N−µ_a^N)δq^a_N = 0;

(λ_p,0^a −ν₀^a)δp⁰_a = 0; (λ_p,N^a −ν_N^a)δp^N_a = 0;

(66)

¯

x(t0) = [0,0,0,0,1,0]^T

¯

p(t0) = [0,0,0,0,0,0]^T

¯

x(tf) = [0,0,0,0.7071,0.7071, ^π₄]^T

¯

p(tf) = [0,0,0,0,0,0]^T (m1,m2) = (1,1)

(l1,l2) = (1,1)

T = 5

N = 100

Table 2. Specified boundary conditions as well as physical and geometric parameters of the mechanical system in the optimal control problem

0 2 4 6

time t -1

-0.5 0 0.5 1

xi

MIN RED

0 2 4 6

time t -1

-0.5 0 0.5 1

pi

MIN RED

0 2 4 6

time t -1

-0.5 0 0.5 1

ui

MIN RED

Figure 4. Solutions of the optimal control problem boundary value problem defined by Table 2. Here the positions (left) and momenta (center) are shown in redundant coordinates while the redundant controls (right) are shown in minimal coordinates.

The boundary conditions chosen for the optimal control example, the length and mass of the slide and the pendulum as well as the length of the time intervalI ∈[0,T]and the number of discrete time intervals,N, is shown in Table 2.

Comparing the solutions depicted in Fig. 4, it can be seen that the two BVP under consideration are indeed equivalent.

Fig. 5 also shows a sequence of the solution of the optimal control boundary value problem.

4 CONCLUSION

In this paper, the boundary value problem of the optimal control problem in redundant coordinates was considered in more detail. Due to the mathematical similarity of the Lagrange formalism of classical mechanics and the optimal control problem, the first step was to consider the boundary

-2 0 2

x-axis -3

-2 -1 0 1 2

y-axis

t = 0

-2 0 2

x-axis -3

-2 -1 0 1 2

y-axis

t = 1.05

-2 0 2

x-axis -3

-2 -1 0 1 2

y-axis

t = 2.1

-2 0 2

x-axis -3

-2 -1 0 1 2

y-axis

t = 3.15

-2 0 2

x-axis -3

-2 -1 0 1 2

y-axis

t = 4.2

-2 0 2

x-axis -3

-2 -1 0 1 2

y-axis

t = 5

Figure 5. Sequence of the motion of the controlled mechanical system defined by the optimal control BVP with the data given in Table 2.

value problem in redundant coordinates in the simplified framework of classical mechanics. After the requirements on the variations on the boundary, which occur due to the algebraic constraints imposed on the redundant coordinates, were discussed on the mechanical level, the knowledge gained from this was transferred to the optimal control problem. Here, the challenging optimal control problem in redundant coordinates was first discussed and the correct necessary optimality conditions were referred to. Subsequently, with the help of the clear separation of the coordinates into normal and tangential parts, which was already known from the mechanical boundary value problem, the optimal control problem was formulated in redundant coordinates. Finally, it was verified by means of a numerical example that the optimal control problem formulated in both redundant and minimal coordinates leads to equivalent numerical results.

ACKNOWLEDGEMENTS

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) – project number 442997215. This support is gratefully acknowledged.

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