The Boltzmann-Hamel Equations for Optimal Control
aa r X i v : . [ m a t h . O C ] J un The Boltzmann-Hamel Equations for OptimalControl
Jared M. Maruskin and Anthony M. Bloch
Department of Mathematics, The University of Michigan
Abstract
We will extend the Boltzmann-Hamel equations to the optimal control setting,producing a set of equations for both kinematic and dynamic nonholonomic optimalcontrol problems. In particular, we will show the dynamic optimal control problemcan be written as a minimal set of 4 n − m first order differential equations ofmotion. Key words:
Nonholonomic Control, Optimal Control, Boltzmann-Hamel equations,quasi-velocities
Quasi-velocity formulations, such as Maggi’s equation and the Boltzmann-Hamel equation, have achieved much success in the analysis of nonholonomicsystems due to their ability to cast the dynamical equations of motion in a
Preprint submitted to Elsevier 24 October 2018 orm requiring fewer equations, see (10), (13), and (11). For an n degree of free-dom system with m nonholonomic constraints, 2 n + m equations of motion arerequired if one uses the fundamental nonholonomic form of Lagranges equa-tion. 2 n differential equations for the system state, and m algebraic relationsthat must be solved for the multipliers. However, if quasi-velocity techniquesare employed, the system can be written as a system of 2 n − m first orderdifferential equations.The standard approach to optimal control problems is to use Lagrange Multi-pliers. Under certain conditions, the optimal control problem can be reformu-lated as a vakonomic (variational nonholonomic) problem (3). One can furtheranalyze optimal control problems with Pontryagin’s Maximum Principle, see(2), (5), or (1). Solutions to the kinematic optimal control problems, whereone has direct control over a number of the velocities, can be expressed using2 n + m equations of motion; whereas solutions to dynamical optimal controlproblems, where one has acceleration controls, can be expressed with 4 n + m equations of motion. Some geometric aspects of this system have been dis-cussed in (6). In this paper, we extend quasi-velocity techniques to optimalcontrol problems with nonholonomic constraints. We show how to write theoptimal control equations for kinematically actuated systems as a system of2 n first order differential equations (a savings of m equations) and the optimalcontrol equations for dynamically actuated systems as a system of 4 n − m first order differential equations (a savings of 3 m equations).2 .2 Summation Convention To aid in notation, we will invoke the summation convention throughoutthis paper. Greek letters ( α, β, γ, . . . ) run over the constrained dimensions1 , . . . , m . Capital letters (
A, B, C, . . . ) run over the unconstrained dimensions m + 1 , . . . , n . Lower case letters ( a, b, c, . . . ) run over all dimensions 1 , . . . , n . In this section we will present the basic background on nonholonomic con-straints and quasi-velocities. We will discuss the basic properties of this con-nection and derive the transpositional relations, (10), (13).
Let Q be the configuration manifold of our system, with dim Q = n and T Q itscorresponding tangent bundle (our phase space). A mechanical
Lagrangian isgiven by L : T Q → R , usually taken to have the form L ( q, ˙ q ) = g ij ˙ q i ˙ q j − V ( q )where g ij is the kinetic energy metric and V ( q ) is a potential term.We further suppose our system is subject to m linear scleronomic (time inde-pendent) nonholonomic constraints, i.e. constraints of the form: a αi ( q ) ˙ q i = 0 (1)Define now a vector space isomorphism Ψ ji on the tangent space, with inversetransformation Φ ij . The first m rows of Ψ ji are taken to agree with the con-straint matrix, i.e. Ψ σi ( q ) = a σi (1). The remaining rows can be choosen freely,3o long as the resulting matrix Ψ is invertible. The transformation Ψ can beviewed as a change of basis:Ψ : ( ∂∂q i ) ni =1 → ( ∂∂θ i ) ni =1 where the new basis is referred to as the quasi-basis . The velocity of the system v ∈ T q Q can be expressed in terms of the ordinary or quasi-basis as follows:˙ q i ∂∂q i = (cid:16) Ψ ji ˙ q i (cid:17) ∂∂θ j = u j ∂∂θ j = (cid:16) Φ ij u j (cid:17) ∂∂q i where the components u j are the quasi-velocities . Basis vectors transform as: ∂∂θ j = Φ ij ∂∂q i and ∂∂q i = Ψ ij ∂∂θ j Finally, one defines a set of n one-forms, dual to the quasi-basis: dθ j = Ψ ji dq i Even though this notation is found in the literature, it is really a notationalmisnomer, as the one forms dθ j are not exact. Definition 1
Consider a curve γ ( t ) : [ a, b ] → Q . A proper variation of γ ( t ) isa differentiable function q ( s, t ) : [ − ε, ε ] × [ a, b ] → Q that satisfies the followingconditions:(i) q (0 , t ) = γ ( t ) , ∀ t ∈ [ a, b ] (ii) q ( s, a ) = γ ( a ) and q ( s, b ) = γ ( b ) , ∀ s ∈ [ − ε, ε ] . Definition 2
The infinitessimal variation δq ( t ) corresponding to the varia-tion q ( s, t ) is the vector field defined along γ ( t ) by δq ( t ) = ∂q ( s, t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =0 .
4e will further assume the variations to be continuous and contemporaneous .Continuity of the variations implies that the Lie Derivative L ˙ q δq ≡ δθ j ( t ) = Ψ ji δq i . A fundamental ingredient for understanding nonholonomic variational prob-lems is the following set of transpositional relations (see (10), (13)).
Theorem 1 (First Transpositional Relations)
Utilizing the shorthand d := ∂/∂t , δ := ∂/∂s , we have: ( dδq i − δdq i )Ψ ji = ( dδθ j − δdθ j ) + γ jab u a δθ b (2) where γ jab are the Hamel coefficients γ spq = ( ∂ Ψ si ∂q j − ∂ Ψ sj ∂q i ) Φ ip Φ jq . The left hand side of (2) is no more than dθ j ( L ˙ q δq ); and, therefore, for con-tinuous variations, is identically zero. We therefore have the following: Corollary 1
For proper, continuous variations, variations of the quasi-velocitiescan be related to variations of the quasi-coordinates as follows: δu j = dδθ j + γ jab u a δθ b (3)Therefore, due to the nonintegrability of the constraint distribution ( γ σij = 0, σ = 1 , . . . , m ), one cannot obtain closure in the quasi-coordinate space, even atthe differential level ((10), (13)). One must choose between δu σ = 0 or dδθ j =0. The correct dynamical equations of motion are obtained if one chooses the5ariations so that they obey the Principle of Virtual Work , δθ j ≡
0. If one,on the other hand, choose the variations to satisfy δu σ = 0, one would obtaintrajectories that satisfy Hamilton’s Principle. Such trajectories are referred toas the vakonomic motion of the system, a term introduced by Arnold. Definition 3
The associated quasi-acceleration, a i , and quasi-jerk, i , are de-fined to be a i = ˙ u i and i = ˙ a i . A direct coordinate calculation shows:
Theorem 2 (Second Transpositional Relation)
For continuous variations,we have δdu i = dδu i . Equivalently, δa i = ∂ ( δu i ) /∂t . We will derive the Boltzmann-Hamel equations for nonholonomic mechanicsdirectly from variational principles. A more algebraic derivation of these equa-tions is given in (10). We will begin with the
Lagrange-D’Alembert Principle : Definition 4 (Lagrange-D’Alembert Principle)
The correct dynamical equa-tions of motion are the ones which minimize the action I = Z ba L ( q, ˙ q ) dt ,where L ( q, ˙ q ) is the unconstrained mechanical Lagrangian and the variationsare chosen to satisfy the Principle of Virtual Work. Let L ( q, u ) = L ( q, ˙ q ( q, u )) be the re-expression of the unconstrained La-grangian in terms of the quasi-velocities. Taking variations of the action andusing the first transpositional relations (3), one obtains:6 I = Z ba ∂ L ∂q i δq i + ∂ L ∂u i δu i + F i δq i ! dt = Z ba ∂ L ∂θ i − ddt ∂ L ∂u i + ∂ L ∂u j γ jki u k + Q i ! δθ i dt where F i is the external applied force and we have defined: ∂ L ∂θ i = ∂ L ∂q j ∂q j ∂θ i = ∂ L ∂q j Φ ji and Q i = Φ ji F j After applying the Principle of Virtual Work, δθ σ ≡
0, the remaining n − m variations δθ I can be taken to be independent, and we obtain the Boltzmann-Hamel equations for nonholonomic mechanics: ddt ∂ L ∂u I − ∂ L ∂θ I − ∂ L ∂u j γ jKI u K = Q I (4)˙ q i = Φ iJ u J (5)One must use the unconstrained Lagrangian for these equations. After thepartial derivatives are taken, one then applies the constraints u σ = 0. TheBoltzmann-Hamel equations (4)- (5) are a minimal set of 2 n − m first orderdifferential equations for the n q i ’s and the n − m u I ’s. In this section we will present a quasi-velocity based method for kinematicoptimal control problems, where one has direct controls over the velocities. Asan example, we will work out the optimal kinematic control equations for thefalling rolling disc. 7 .1 Theory
For a general affine kinematic control system subject to m nonholonomic con-straints, the following system is typically specified: ˙ q i = X iI ( q ) w I , where the w I are the n − m controls and X iI ( q ) is the i -th component of the I -th independentcontrol vector field. Taking the m constraints as the first m quasi-velocities: u σ = Ψ σi ˙ q i ≡ , (6)one can, wlog, take the controls as the remaining independent quasi-velocities: w I ( q, ˙ q ) = u I = Ψ Ii ˙ q i (7)With this choice, the control vector fields are thus identifies with the last n − m columns of Φ = Ψ − , i.e. X iI = Φ iI .For a given cost integrand g ( q, w ), the Kinematic Optimal Control Problemis then given by minimizing the cost function I = Z ba g ( q, w ) dt over all curvessatisfying (7)-(6) with fixed endpoints q ( a ) and q ( b ).We now define the quasi-basis so that Ψ σi = a σi , as usual, and, additionally,so that Ψ Ii = b Ii . Then the constraints can be written u σ , and the n − m control variables w I coincide with the remaining n − m free quasi-velocities u I . Define now C ( q, u ) = g ( q, w ( q, ˙ q ( q, u ))). In our case, we have chosen theunconstrained quasi-velocities to coincide with the controls, i.e. u I = w I , thuswe will have C ( q, u ) = g ( q, u ).In order to enforce (6), we must apply the Lagrange Multipliers to the costfunction before taking variations. In this case, we are selecting Hamilton’s Prin-ciple, where the cost function is minimized amongst the set of kinematically8dmissable curves. We then take unconstrained variations of the augmentedcost function I = Z ba ( C ( q, u ) + µ σ u σ ) dt . Since C ( q, u ) only depends on theunconstrained quasi-velocities u I , we have: δI = Z ba ∂C∂θ i δθ i + ∂C∂u I δu I + µ σ δu σ + u σ δµ σ ! dt Setting the coefficients of the δµ σ terms returns our constraints u σ = 0. Leav-ing this term off for now, using the transpositional relations (3), and integrat-ing by parts yields δI = Z ba ( ∂C∂θ i + ∂C∂u I γ Isi u s + µ σ γ σsi u s ! δθ i − ddt ∂C∂u I δθ I − ˙ µ σ δθ σ ) dt We thus have the following
Theorem 3
The Boltzmann-Hamel equations for the kinematic optimal con-trol problem are: ddt ∂C∂u I − ∂C∂θ I − ∂C∂u J γ JSI u S = µ τ γ τSI u S (8) − ∂C∂θ σ − ∂C∂u J γ JSσ u S = − ˙ µ σ + µ τ γ τSσ u S (9)˙ q i = Φ iS u S (10)These represent a minimal set of 2 n first order differential equations: the n − m equations (8) for the unconstrained u I ’s, the m equations (9) for the multipliers µ σ ’s, and n kinematic relations (10) for the q i ’s.As an interesting aside, if the cost function integrand C ( q, u ), when expressedin terms of the quasi-velocities, is identical to the constrained mechanicalLagrangian, then these equations produce the vakonomic motion associatedwith the system. See (3) for additional discussion on the coincidence of thevakonomic motion (Lagrange’s Problem) and the optimal control problem.9 .2 Optimal Control of the Heisenberg System The optimal control of the Heisenberg system, discussed in (4) and (2), isa classical underactuated kinematic control problem. Local coordinates aregiven by q = h x, y, z i . For this system, one has velocity controls w = ˙ x and w = ˙ y and the motion is subject to the nonholonomic constraint ˙ z = y ˙ x − x ˙ y .The control velocity field is therefore given by:˙ q = X w + X w , where X = h , , y i T and X = h , , − x i T . Using these controls, one seeks tosteer the particle from the point h , , i at time t = 0 to the point h , , a i attime T >
0, while minimizing the functional I = 12 Z T (cid:16) w + w (cid:17) dt .We will derive the equations of motion which yield this solution path viathe vakonomic form of the Boltzmann-Hamel equations. We choose quasi-velocities: u = y ˙ x − x ˙ y − ˙ z , u = ˙ x , and u = ˙ y . Notice the quasi-velocities u and u coincide with the control velocities. The transformation matrices Ψand Φ are given by:Ψ = y − x −
11 0 00 1 0 and Φ = − y − x The nonzero Hamel coefficients are γ = − γ = 2. Expressing the integrandof the cost function in terms of quasi-velocities yields C = 12 (cid:16) u + u (cid:17) . Thekinematic optimal control Boltzmann-Hamel equations (8)-(10) immediatelyproduce the following set of first order differential equations:˙ x = u ˙ y = u ˙ z = − u + yu − xu ˙ u = − µu ˙ u = 2 µu ˙ µ = 010here µ ( t ) = µ (0) is an arbitrary constant that can be choosen such that thesolution trajectory reaches its final destination point. The top equations area reiteration of the control field ˙ q = X w + X w = X u + X u and thebottom equations produce the optimal control. The generalized coordinates of the vertical rolling disc are given by q = h x, y, θ, φ i , where ( x, y ) is the contact point of the disc and the x − y plane, φ is the angle the disc makes with the x -axis, and φ is the angle a referencepoint on the disc makes with the vertical. Assume we have the kinematiccontrols w = ˙ θ and w = ˙ φ , and that the motion is subject to the nonholo-nomic constraints ˙ x − cos( φ ) ˙ θ = 0 and ˙ y − sin( φ ) ˙ θ = 0. This gives rise tothe control vector field ˙ q = X w + X w where X = h cos φ, sin φ, , i T and X = h , , , i T .We wish to steer the disc between two points while minimizing the costfunctional 12 Z ba ( w + w ) dt . We choose quasi-velocities u = ˙ x − cos( φ ) ˙ θ , u = ˙ y − sin( φ ) ˙ θ , u = ˙ θ , and u = ˙ φ , so that the transformation matrices Ψand Φ are given by:Ψ = − cos φ
00 1 − sin φ
00 0 1 00 0 0 1
Φ = φ
00 1 sin φ
00 0 1 00 0 0 1
The Hamel coefficients are: γ = sin φ = − γ and γ = − cos φ = − γ .In terms of the quasi-velocities, the integrand of the cost function becomes C ( q, u ) = u + u . The Boltzmann-Hamel equations (8)-(10) then producethe following set of first order differential equations:11 u = ( µ cos φ − µ sin φ ) u ˙ µ = 0 ˙ x = cos( φ ) u ˙ θ = u ˙ u = ( µ sin φ − µ cos φ ) u ˙ µ = 0 ˙ y = sin( φ ) u ˙ φ = u The falling rolling disc can be described by the contact point ( x, y ) and Clas-sical Euler angles ( φ, θ, ψ ), as shown in Figure 1. We will take the coordinateordering ( φ, θ, ψ, x, y ).PSfrag replacements e d ˙ φ ˙ θ ˙ ψ e θ e ψ PCrx yzφ ψθ
Fig. 1. Euler Angles of the Falling Rolling Disc
Suppose we have direct control over the body-axis angular velocities w = ω d := ˙ φ sin θ , w = ˙ θ , and w = Ω := ˙ φ cos θ + ˙ ψ (in the e d , e θ , ande ψ directions, respectively (see Fig. 1)), and the system is subject to thenonholonomic constraints ˙ x + r ˙ ψ cos φ = 0 and ˙ y + r ˙ ψ sin φ = 0. We wishto steer the disc between two points while minimizing the cost functional I [ γ ] = 12 Z ba (cid:16) w + w + w (cid:17) dt . We will choose as quasi-velocities u = ˙ φ sin θ , u = ˙ θ , u = ˙ φ cos θ + ˙ ψ , u = ˙ x + r ˙ ψ cos φ , and u = ˙ y + r ˙ ψ sin φ . The quasi-velocities ( u , u , u ) = ( ω d , ˙ θ, Ω) represent the angular velocity expressed inthe body-fixed frame, and are coincident with the kinematic controls. Theseare not true velocities (like the Euler Angle Rates), as they are non-integrable.The nonholonomic constraints in terms of these variables are u = u = 0.12he transformation matrices areΨ = sin θ θ r cos φ r sin φ and Φ = csc θ − cot θ r cos φ cot θ − r cos φ r sin φ cot θ − r sin φ The nonzero Hamel-coefficients are γ = − cot θ = − γ , γ = 1 = − γ , γ = r sin φ csc θ = − γ , and γ = − r cos φ csc θ = − γ .Written in terms of the quasi-velocities, the integrand of the cost function is C ( q, u ) = ( u + u + u ). The kinematic optimal control Boltzmann-Hamelequations (8)-(10) give us a minimal set of 10 first order differential equations:˙ u = u u − u u cot θ − r ( µ sin φ − µ cos φ ) csc θu ˙ u = u cot θ − u u ˙ u = r ( µ sin φ − µ cos φ ) csc θu ˙ µ = 0 , ˙ µ = 0˙ φ = csc θu , ˙ θ = u , ˙ ψ = − cot θu + u ˙ x = r cos φ cos θu − r cos φu , ˙ y = r sin φ cot θu − r sin φu In this section, we will derive a set of Boltzmann-Hamel equations for thedynamic optimal control problem, which is normally a fourth order system.We will present a minimal set of 4 n − m first order differential equations thatproduces the optimal control, and then discuss examples.13 .1 Boltzmann-Hamel Equations for Optimal Dynamic Control Given a nonholonomic mechanical system with n − m independent accelerationcontrols, it can be recast into the form given by the dynamical Boltzmann-Hamel equations (4)-(5). The dynamical optimal control problem is the prob-lem of finding solution curves between two fixed points h q ( a ) , ˙ q ( a ) i and h q ( b ) , ˙ q ( b ) i that minimize the cost function I = Z ba g ( q, ˙ q, Q ) dt . Utilizing (4) and (5), wecan rewrite the integrand as an explicit function of the coordinates, quasi-velocities, and quasi-accelerations C ( q, u, a ) = g ( q, ˙ q ( q, u ) , Q ( q, u, a )).Since the Boltzmann-Hamel equations no longer depend on the constrainedquasi-velocities and quasi-accelerations, C ( q, u, a ) is also independent of u σ and a σ . Taking variations yields: δI = Z ( ∂C∂q i δq i + ∂C∂u J δu J + ∂C∂a J δa J ) dt .Using the second transpositional relations Theorem 2 for δa J and then inte-grating by parts we obtain δI = Z ( ∂C∂q i δq i + " ∂C∂u J − ddt ∂C∂a J δu J ) dt . Defin-ing the parameters κ J = ∂C∂u J − ddt ∂C∂a J (11)and using the first Transpositional relations (3) we obtain: δI = Z ( ∂C∂θ r − ˙ κ J δ Jr + κ J γ Jsr u s ) δθ r dt These variations are not free, but subject to the nonholonomic constraints a σi ˙ q i = 0. We form the augmented cost integrand by replacing C ( q, u, a )with C ( q, u, a ) + µ σ u σ . Taking variations, the δµ σ coefficients recover the con-straints. Ignoring these terms, we are left with δI = Z ( ∂C∂θ r − ˙ κ J δ Jr + κ J γ Jsr u s − ˙ µ σ δ σr + µ σ γ σsr u s ) δθ r dt where the variations are now taken to be unconstrained. Notice the multipliers14 σ are not the mechanical multipliers, but a multiplier on the cost functionthat enforces Hamilton’s Principle. We thus have the following: Theorem 4
The Boltzmann-Hamel equations for Optimal Dynamic Controlare given by: − ∂C∂θ A + ˙ κ A − κ J γ JSA u S = µ τ γ τSA u S (12) − ∂C∂θ σ − κ J γ JSσ u S = µ τ γ τSσ u S − ˙ µ σ (13)˙ q i = Φ iS u S (14)The optimal control system can therefore be given by a minimal set of 4 n − m first order differential equations as follows. We have n kinematic relations(14), 2 n − m relations ˙ u A = a A and ˙ a A = A , n − m equations for ˙ A (given by inserting (11) into (12)), and, finally, m relations for the multi-pliers ˙ µ σ (13). Once the resulting optimal control dynamics are determined,the control forces which produce the optimal trajectory are then given by the n − m algebraic equations (4). The solution is then found by solving the re-lated boundary value problem, with 4 n − m prescribed boundary conditions: q i (0) , u A (0) , q i ( T ) , u A ( T ). Consider the vertical rolling disc of § θ and φ directions. The corresponding dynamical equations of motion (see (2)) are: ¨ θ = w , ¨ φ = w , ˙ x = ˙ θ cos φ , and ˙ y = ˙ θ sin φ . This is equivalent to a minimalset of 6 first order differential equations (the number obtained by using theBoltzmann-Hamel equations (4) and (5).15e now wish to choose the control forces so as to minimize the cost function R ( w + w ) dt . Solving for the controls in terms of the quasi-accelerations w = ¨ θ = a and w = ¨ φ = ¨ a , this is equivalent to minimizing theaction Z (cid:18) a + 132 a (cid:19) dt subject to the nonholonomic constraints. Using thedynamic optimal control Boltzmann-Hamel equations (12) and (13), coupledwith the dynamical equations of motion above, and eliminating the controls,we have a minimal system of 12 first order differential equations:˙ x = cos φ u ˙ = 49 ( µ sin φ − µ cos φ ) u ˙ y = sin φ u ˙ = 16( − µ sin φ + µ cos φ ) u ˙ θ = u ˙ u = a ˙ a = ˙ µ = 0˙ φ = u ˙ u = a ˙ a = ˙ µ = 0By use of quasi-velocities, quasi-accelerations, and quasi-jerks, we have madethe following simplifications: u = u = a = a = = = 0, thereby elimi-nating the necessity of 6 of the 18 first order differential equations necessaryin the standard approach. The solution to this system of differential equationsyields the optimal dynamic control equations of the vertical rolling disc. It isequivalent to the following reduced system˙ x = cos φ ˙ θ ˙ y = sin φ ˙ θ .... θ = 49 ( µ sin φ − µ cos φ ) ˙ φ .... φ = 16( − µ sin φ + µ cos φ ) ˙ θ where µ , µ are constants. Consider dynamic control of the free rigid body, where the generalized coordi-nates are given by the Type-I Euler angles ( ψ, θ, φ ). As quasi-velocities, choose16he body-fixed components of the angular momentum u = ω x = − ˙ ψ sin θ + ˙ φ , u = ω y = ˙ ψ cos θ sin φ + ˙ θ cos φ , and u = ω z = ˙ ψ cos θ cos φ − ˙ θ sin φ . Thetransformation matrices are given as:Ψ = − sin θ θ sin φ cos φ θ cos φ − sin φ and Φ = θ sin φ sec θ cos φ φ − sin φ θ sin φ tan θ cos φ The mechanical Lagrangian is given as L ( q, u ) = 12 ( I xx u + I yy u + I zz u ).The nonzero Hamel coefficients are γ = 1, γ = − γ = 1, γ = − γ = 1, and γ = −
1. For notational convenience, define η = I zz − I yy , η = I xx − I zz , and η = I yy − I xx . Then the Boltzmann-Hamel equations (4)produce the Euler Equations: I xx ˙ u + η u u = M x I yy ˙ u + η u u = M y I zz ˙ u + η u u = M z (15)where M x , M y , and M z are the control torques applied about the body fixedprincipal axes. The cost function integrand ( M x + M y + M z ), when ex-pressed in terms of quasi-variables, is given by: C = { I xx a + I yy a + I zz a +2 I xx η a u u + 2 I yy η u a u + 2 I zz η u u a + η u u + η u u + η u u } .The κ ’s (11) are given by: κ = I yy η a u + I zz η u a + η u u + η u u (16) − I xx − I xx η u a − I xx η a u κ = I xx η a u + I zz η u a + η u u + η u u (17) − I yy − η I yy u a − η I yy a u κ = I xx η a u + I yy η u a + η u u + η u u (18) − I zz − η I zz u a − η I zz a u The optimal control Boltzmann-Hamel equations (12) then work out to be:˙ κ = κ × ω (19)These provide 3 differential equations for the ˙ ’s. Let I be the moment inertia17ensor with respect to the principal axes basis ˆ e x , ˆ e y , ˆ e z , so that, in dyadicnotation, I = I xx ˆ e x ˆ e x + I yy ˆ e y ˆ e y + I zz ˆ e z ˆ e z . Let Π := I · ω be the body axisangular momentum, and κ = h κ , κ , κ i . Then (16)-(18) can alternatively bere-expressed as: κ = Π × ˙ Π + Π × ( ω × Π ) − ¨ Π − I · n ω × ˙ Π + ˙ ω × Π + ω × ( ω × Π ) o (20)Finally, by defining λ ( ω , ˙ ω ) = κ + ¨ Π , the dynamic optimal control equationsfor the free rigid body can be expressed as:... Π = ˙ λ + ¨ Π × ω − λ × ω (21)In addition, we have the kinematic relations˙ ψ = sec θ sin φu + sec θ cos φu (22)˙ θ = cos φu − sin φu (23)˙ φ = u + tan θ sin φu + tan θ cos φu (24)as well as the relations ˙ u i = a i , ˙ a i = i . This is a set of 12 first order differentialequations. Once one solves the corresponding boundary value problem (initial,final Euler angles, angular velocities specified), the controls are determins bythe algebraic relations (15).For the special case when the rigid body is spherical one sees from (20) that κ = − ¨ Π and λ = . Then the Boltzmann-Hamel equations for the optimaldynamic control of the free rigid body (21) reduce to ... ω = ¨ ω × ω . When coupledwith the kinematic relations (22)-(24) and the algebraic relations (15), theoptimal control trajectories of the free rigid sphere are produced. Integratingonce yields the second order system ¨ ω = c + ˙ ω × ω , which coincides with theresult of (12). See also (8). The optimal solution trajectory of the reorientationof the rigid sphere from q (0) = h , , i , ω (0) = h , , i to the point q (1) =18 PSfrag replacements ψ θ φ ω x ω y ω z Fig. 2. Optimal Dynamic Control of Free Sphere: Euler Angles and Body FixedAngular Velocity with respect to time. h π, − π/ , π/ i , ω (1) = h , , i is plotted in Fig. 2. In this paper, we showed how one can extend quasi-velocity techniques tokinematic and optimal control problems. Standard Lagrange Multiplier tech-niques for kinematical optimal control problems produce a set of 2 n + m firstorder differential equations: n for the coordinates q i , n for the velocities ˙ q i ,and m for the multipliers µ σ . On the other hand, by generalizing the dynamicBoltzmann-Hamel equations to the kinematic control setting (Theorem 3), weobtain a savings of m first order differential equations, as one no longer needsolve for the constrained quasi-velocities. Moreover, the differential equationsfor the multipliers (9) are naturally separated from the differential equationsfor the quasi-velocities (8).For the dynamic optimal control problem, one typically encounters a fourthorder system, plus multipliers, which produces a total of 4 n + m first order19ifferential equations. The Boltzmann-Hamel form of the equations (Theorem4) gives a minimal set of 4 n − m equations of motion, as one no longer needintegrate the m constrained quasi-velocities, quasi-accelerations, and quasi-jerks, u σ ≡ a σ ≡ σ ≡
0, respectively. This approach gives us a totalsavings of 3 m first order differential equations. Initial and final conditions arethen enforced by solving the resulting system of differential equations as a twopoint boundary value problem.The authors wish to thank support from NSF grants DMS-0604307 and CMS-0408542. References [1] A. Agrachev and Y. Sachkov,
Control Theory from the Geometric Viewpoint , Springer, 2004.[2] A.M. Bloch,
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