Generalized Momenta in Constrained Non-Holonomic Systems–Another Perspective on the Canonical Equations of Motion Alan A. Barhorst∗ Professor Department of Mechanical Engineering Texas Tech University Lubbock, Texas 79409-1021 Email:
[email protected]
In this paper generalized momenta are defined for nonholonomic dynamical systems in a possibly novel formalism. The momenta are defined for Lagrangian/energy systems as well as projective d’Alembert systems, thus allowing canonical equations to be developed. The momenta can be defined in terms of quasi-variables (generalized speeds) or with ultimately an excess of the time derivatives of the generalized coordinates. This work may lay a framework to build nonholonomic generalizations to the Hamilton-Jacobi canonical transformation equations. In the sense that these equations are fundamental and minimalistic, they too can be considered canonical in their nature. For immediate practical consideration, demonstrated is significant reductions of the time required to numerically integrate the models formulated with these techniques, indicating that this modeling formalism may be more than an intellectual curiosity.
1
Introduction In previous work a formulation of the equations of motion for non-holonomic systems was presented that cast the problem in a form that resembles state feedback control laws [1]. In this paper the Moore-Penrose pseudo-inverse is used to write the control-like equations, so are a least-squaresfit-like model of the constrained motion. In previous work motivating the work herein, the author developed the same control-like equations but exactly satisfying the constraints of the system not requiring the least squares fit of the pseudoinverse [2]. Later this work was revisited by others to demonstrate utility [3]. The work presented in [2] provides the foundation for the methodology delineated below. The foundation of the work herein is also supported by the works of many others [4,5,6,7,8], and the previous work of the author in elasto-dynamic modeling [9, 10, 11, 12, 13, 14]. In a comprehensive review paper [15], the author of the
∗ Address
all correspondence for other issues to this author.
paper summarizes the current state of the science for modeling discrete coordinate mechanical systems. The review summarizes all the published work over the last several hundred years of the development of analytical mechanics. In this work arguments are made stating there is no need to present “new” methods to formulate equations of motion, under the subspecialties of multi-body dynamics and other newly named sub-disciplines of analytical mechanics, because the foundations are already completely developed in the work the author presents in review. However others have not heeded this suggestion and proceeded to developed new ideas in modeling rigid body mechanical systems. Two papers of note describe formulations of combined holonomically and nonholomically constrained mechanical systems as first order ordinary differential equations implicitly satisfying all constraints [16, 17]. In [18] the work in [15] is formalized into a complete textbook treaties on Analytical Mechanics with an expos´e on constrained system dynamics. In chapter 8.2 of this work, non-holonomic generalized momenta are defined from Hamilton’s Canonical Equations in terms of quasi-variables and thus are a reaction-less (i.e. free of Lagrange multipliers) model of the motion. The author attributes this work and its applications to others appearing in the literature spanning the time frame 1913-1989 infrequently and does not pursue the work further in the book. As with all work in analytical mechanics one wonders if possible novel ideas have been explored in the storied past of these topics of practical and theoretical interest. Sometimes a possibly naive look at these centuries old ideas can shed some new light that may lead to new insights or simply remain an intellectual exercise in good faith. In what follows, in this introductory section, we present two of the current formulations for non-holonomic systems (as reviewed in [4, 15]), that can be considered canonical in the sense that they are fundamental standard first order formulations, but may not be canonical in the sense the word
is used in the Hamilton-Jacobi transformation theory [4], where the symmetry of the model allows certain geometric transformations to uncover integrals of the motion. The remainder of this introduction motivates the development that follows. We do not present in review the canonical formalism presented in [18] because it was not known to the author of this work over the course of his on-again-off-again studies of dynamic modeling from 1986 through the present time, some of it cited above, that lead to the development below. The author will refer the reader to the referenced work [18] and beg their indulgence in the development that follows, see the footnote 2 at equation 7 for comments regarding the novelty of the work presented herein relative to the work cited in [18]. 1.1
Hamilton’s Canonical Equations for Nonholonomic Systems The Hamilton-Jacobi transformation theory of differential equations is developed from the canonical equations of motion of Hamilton [4] and are postulated for holonomic systems with these constraints satisfied by the choice of independent generalized coordinates qi , i = 1, 2, ..., N of the holonomic configuration space. From the method of d’Alembert and Lagrange, define the Lagrangian as L = T − V , kinetic energy minus potential energy, then the N coupled second order equations of motion are given by: d dt
∂L ∂q˙i
−
∂L = Qi + Q0i , ∂qi
i = 1, 2, 3, ..., N
(1)
Where the polygenic1 holonomic generalized force/torque can be computed from: ∂~v j ~ ∂~ω j Qi = ~Fj · +Mj · ∂q˙i ∂q˙i
(2)
where the implied sum on repeated index j is taken over all bodies, and the forces and moments are resultants of all active forces acting on the jth body at (for convenience) the center of mass moving at velocity ~v j and the jth body rotates with angular velocity ~ω j . The non-holonomic generalized force Q0 is defined for M constraints of the form:
Nk = aki q˙i = 0,
sum on i = 1, 2, ..., N, for each k = 1, 2, ..., M
(3)
where Lagrange multipliers λk are utilized making equations 1 and 3 differential-algebraic second order ordinary differential equations (DAE). The coefficients of the q¨i are functions of the qi in general thus require significant algebra, symbolic or numeric, to be completed before quadrature ensues to find qi (t). Equations 1 through 4 are Lagrange’s equations for a non-holonomically constrained system. Defining the generalized momenta as: pi =
∂L ∂q˙i
(5)
solvable for the q˙i in terms of the pi , and the Hamiltonian as: H = pi q˙i − L,
sum on i = 1, 2, ..., N
(6)
Legendre’s dual transformation [4] permits the canonical equations of motion for a non-holonomic system to be written as2 : p˙i = −
∂H + Qi + Q0i ∂qi (7)
∂H q˙i = ∂pi where q˙i has been eliminated from H via equation 5. These 2N first order differential equations and the M constraints from equation 3 can be solved simultaneously for the motion of the system and the constraint Lagrange multipliers. The formal Hamilton-Jacobi theory of continuous differential transformations as a solution to equations 7 only apply for monogenic forces thus do not include the polygenic generalized forces Qi and Q0i . In the literature, the canonical equations are those for which the Hamilton-Jacobi transformation theory applies, so do not encompass nonholonomically constrained systems. So in the strictest sense (i.e. simplest formulation), equations 7 are the canonical equations for a non-holonomic system, they are 2N + M differential-algebraic equations in 2N + M unknowns qi , pi and λk . They must be solved as DAE. A point to note is that the 2N dimensional phase space spanned by qi and pi is bounded inherently by the constraints in equations 3, therefore the multipliers λk can be thought of as penalty parameters (forces of constraint) that must be found as functions of time to keep the evolution of the momenta and configuration coordinates in the phase space bounded by the constraint surfaces.
as: Q0i = λk aki ,
sum on k = 1, 2, ..., M for each i
(4)
1 We borrow this terminology from Lanzcos [4] and others where monogenic forces are derivable from a single scalar work function and polygenic forces are not so derivable (i.e. conservative and non-conservative forces, respectively).
2 In [18], Ch 8.2, these equations are written as 2N − M equations in 2N unknowns where Q0i has been absorbed into a non-holonomic Hamiltonian H ∗ by the definition of quasi-variables shown in equation 8 and appropriate adjustments made to the Lagrangian/Hamiltonian partial derivatives with respect to the quasi-variables. The final form of the first of equations 7 in [18] is as shown above but generalized coordinates replaced with quasiccordinates and with Q0i replaced with a term that sums selected products of generalized momenta with the gradient of the non-holonomic kinetic energy with respect to the non-holonomic momenta. M addtional non-holonomic equations of the form 3 or 10 are also required for quadrature.
1.2
Projective Dynamical Equations for Nonholomomic Systems From this perspective d’Alembert-Lagrange principle is kept in vector form and is a generalization of the principle of virtual work. This is the general approach of Gibbs, Appell, Kane, Maggi, [5, 6, 7, 19], and others [15, 18, 20]. For the holonomic configuration space, define N independent generalized coordinates qi , i = 1, 2, ..., N. Further define N generalized speeds or quasi-coordinates as: ui = Yi j q˙ j + Zi ,
sum on j = 1, 2, ..., N for each i = 1, 2, ..., N
(8)
where Yi j = Yi j (qk ) and Zi = Zi (qk ). Insist that we can write the inverse as:
where summation of j over all bodies is implied for each ~ j are resultant noni ∈ I . The forces ~Fj and moments M constraint (active) forces and their moments (plus couples) acting at the center of mass of the jth body. This results in a set of N − M ordinary differential equations of the form: {u˙i } = [I(qk )]−1 {hi (ui , qk , f j )}
where f j are any number of inputs and the inertia matrix [I(qk )] is positive definite. Accompanying this are the N kinematic ordinary differential equations: q˙ j = (K ji + K jd Adi )ui
q˙i = Ki j u j + Ji ,
sum on j = 1, 2, ..., N for each i = 1, 2, ..., N
sum on i = 1, 2, ..., N for each k = 1, 2, ..., M
(10)
Choose M dependent speeds ud and write them in terms of N − M independent speeds via equations 10 as ud = Adi ui ,
sum on i ∈ I for each d ∈ D
(11)
where I is the set of N − M indices associated to independent speeds and where D is the set of M indices associated to dependent speeds. Define the translational momentum of the jth body of mass m j as: ~L j = m j~v j
(12)
where the velocity of the center of mass (used for convenience of formulation) ~v j = ~v j (ui , qk ) with i ∈ I and k = 1, 2, ..., N. Also define the angular momentum of the jth body with central mass moment of inertia dyadic~~I j as: ~ j =~~I j · ~ω j H
(16)
(9)
where Yi j K jk = δik (sum on j, Kroneker delta), and Ji = −Kik Zk (sum on k). Write M non-holonomic constraints or conditions3 as:
Nk = aki ui = 0,
(15)
(13)
where ~ω j = ~ω j (ui , qk ) with i ∈ I and k = 1, 2, ..., N. The equations of motion are given by: ∂~ω j ∂~v j ~ d ~ d ~ ~ · Fj − Lj + · Mj − Hj (14) 0= ∂ui dt ∂ui dt
3 Any transcendental holonomic condition between the generalized coordinates can be differentiated to create a constraint of this same form.
for each j = 1, 2, ..., N, sum on d ∈ D and i ∈ I . Equations 15 and 16 are 2N − M purely differential equations of motion. They are non-DAE and smaller in size than the Hamilton canonical form shown in equations 7, but do require the inverse of an N − M square matrix before quadrature commences. Instead of the phase space this motion can be visualized in the q-configuration space [4] as evolution of the c-point along a curve that at every time has a tangent subspace where the u’s evolve, and a differential mapping of the u’s into the q-space. In [15] equations of the form 16 (or similar) are called reaction-less models, herein we refer to these as “canonical” in the sense that are the simplest set of non-DAE equations of motion required to completely evolve the motion of the modeled system, but not canonical in the sense of HamiltonJacobi integration of equation system 7 in the polygenic force-free case. 1.3
Summary and Motivation It is clear that these approaches provide advantages and disadvantages. In the case of Hamilton canonical equations for non-holonomic systems, a set of DAE result. Advantages are that the phase space retains its full 2N size but is bounded by surfaces where the constraints hold true. Another disadvantage is that the penalties for violating constraints must be determined as part of the solution. Since the generalized forces break the essential symmetry of the equations, the Hamilton-Jacobi theory is not applicable so straight DAE quadrature is required. In the case of the projective equations, the advantage is a smaller set of non-DAE equations result, but algebraic inversion of the inertia matrix is required. As well, depending on parameterization, the kinematic equations can have singular points, unless Euler parameters or similar are used to parameterize the direction cosines. So to motivate the approaches to be presented below, we desire to eliminate the DAE nature of the associated canonical equations and reduce the requirement to invert the inertia matrix while avoiding pitfalls associated to singularities in kinematic differential equations.
In what follows we present the derivation of a “canonical” form for non-holonomic systems from the perspective of the projective d’Alembert form and the LagrangeHamitonian form of the equations of motion. We follow this with expository and numerical examples.
2
Canonical Equations for Non-Holonomic Systems In this section we will derive generalized momentum for each of the projective and Hamiltonian systems and develop canonical equations for the systems. 2.1
Projective Method We start with the d’Alembert-Lagrange variational principle written for rigid body systems [12]: d ~ ~ d ~ ~ ~ Lj · δ~r j + M j − Hj · δθ j 0 = Fj − dt dt
(17)
(18)
∂~r
where we use ~v j = ∂qji q˙i + ∂tj for the last equality. Summation on i = 1, 2, ..., N, is implied. The virtual rotation ~ j = δθ ~ j (δqi ), in the same way that ~ω j = ~ω j (q˙i ), so we δθ can write: ~ j = ∂~ω j δqi δθ ∂q˙i
∂~v j δuk ∂uk
~ j = ∂~ω j δuk and δθ ∂uk
(23)
Using this result, the variational principle 17 can be written as: ∂~v j ~Fj − d ~L j · δuk dt ∂uk ∂~ω j ~j ~ j− d H · δuk + M dt ∂uk
(24)
summation over all bodies j and sum k = 1, 2, ..., N. These variations are not independent due to nonholonomic constraints 10. Varying equation 10 on the u’s only allows us define dependent and independent variations as: sum on i ∈ I for each d ∈ D
(25)
So the variational principle 24 becomes: ∂~v j ∂~v j d ~ ~ 0 = Fj − Lj Adi δui · + dt ∂ui ∂ud ∂~ω j ∂~ω j d ~ ~ + Mj − Hj Adi δui (26) · + dt ∂ui ∂ud
(19) ∂~v j ∂~v j d ~ ~ 0 = Fj − + Lj · Adi dt ∂ui ∂ud ∂~ω j ∂~ω j d ~ ~ + Mj − Hj · + Adi dt ∂ui ∂ud
(27)
(20)
This is N − M ordinary differential equations in 2N unknowns (qk , uk ), We supplement with the M equations 11, and the N kinematic differential equations 16. In this form of the equations of motion we can clearly see the contribution of the projection vector that is due to the constraint, that is the d’Alembert force and torque deficits projected along ∂~ω j ∂~v j ∂ud Adi and ∂ud Adi . We continue, define generalized momenta:
(21)
~ j ·W ~ jic pi = ~L j · ~V jic + H
summation on k implied. Similarly we can write: ∂~ω j ∂~ω j = Yki ∂q˙i ∂uk
δ~r j =
These are independent and arbitrary variations so the equations of motion are for each i ∈ I :
summation on i implied. For the quasi-coordinates or generalized speeds introduced in equations 8 and 9 we can write: ∂~v j ∂~v j ∂uk = ∂q˙i ∂uk ∂q˙i ∂~v j = Yki ∂uk
(22)
Using equations 20-22 in equations 18 and 19 and the fact that Yki Kik = δkk = 1 (Kronecker delta) gives:
δud = Adi δui ,
δ~r j =
∂~r
δqi = Kik δuk
0=
where we use the terms from above as well as the virtual ~ j . The virtual displacement δ~r j and virtual vector rotation δθ rotation can be formed by successive simple virtual Euler rotations in the same way angular velocity can be constructed via Euler rotations. The position vector is ~r j =~r j (qi ,t) allowing: ∂~r j δqi ∂qi ∂~v j δqi = ∂q˙i
Any independent variation can be written as a linear combination of another set of independent variations. Choosing the same coupling as equation 9 we write:
(28)
where the vectors of constraint concordance are given by: ~V jic =
∂~v j ∂~v j + Adi ∂ui ∂ud
∂~ω j ∂~ω j + Adi ∂ui ∂ud
(30)
Equation 28 is linear in the ui , i ∈ I and can be solved as: ui = Bi j p j .
(31)
where Bi j = Bi j (qk ), k = 1, 2, ..., N and is positive-definite. Equation 27 becomes: d pi = Qi + Pi dt
(32)
~ j ·W ~ jic Qi = ~Fj · ~V jic + M
(33)
where:
are the generalized forces and are functions of the applied non-constraint forces and torques, the generalized coordinates qk , and possibly speeds uk 4 . Also where: d ~ c ~ d ~ c Pi = ~L j · V ji + H j · W ji dt dt
(34)
are the projections of the translational and angular momenta along the rate of change of the vectors of constraint concordance. The right-hand side of equation 34 is a mixed function of (qk , q˙k , uk ) bi-linear in (q˙k , uk ) and needs to be written in terms of (qk , pi ). Define: ∂Pi ∂Pi q˙ j + uk (35) ∂q˙ j ∂uk ∂Pi ∂Pi ∂Pi = (K jl + K jd Adl )ul + ul + Adl ul , l ∈ I ∂q˙ j ∂ul ∂ud ∂Pi ∂Pi ∂Pi = (K jl + K jd Adl ) + + Adl Bli pi ∂q˙ j ∂ul ∂ud = Ri (qk , pi ), k = 1, 2, ..., N, and i ∈ I
Ri =
where equations 9, 10, and 31 were used. So the 2N − M canonical equations are: p˙i = Qi + Ri
4 Any
31.
q˙ j = (K jk + K jd Adk )Bki pi
(29)
and ~ jic = W
and
(36)
speeds in Qi need to be written in terms of pi via equations 11 and
(37)
where i, k ∈ I and j = 1, 2, ..., N. These are ordinary differential equations. Comparing equations 36 and 37 to equations 7 (also see footnote 2 near equations 7) we see that we have created a non-DAE form of the canonical equations. They do not have the symmetry of the Hamiltonian function but eliminate the need for penalty constraint Lagrange multipliers. It is easier to prescribe forcing that conserves generalized momenta, but does not lend itself to ignorable coordinates in the absence of generalized forces. Comparing equations 36 and 37 to equations 15 and 16 we see that the number of equations are the same but the need to invert the inertia matrix is not present. In the previous equations it is hard to discern when momenta are conserved, but as stated above, much easier with the new equations. Singularities in the kinematic mapping may be washed out with the coupling of the derivatives of the generalized coordinates to the momenta instead of the generalized speeds. Forming the mapping Bi j in equation 31 does require the inversion of an N − M square positive definite coefficient matrix that is a function of the qk , but requires less effort because the terms arise at the velocity level and thus are simpler to manipulate. This also holds for the Hamilton canonical equations when solving for the q’s ˙ in terms of the p’s at equation 5. 2.1.1
Comments on Momentum and Energy Conservation We can see that pi = const if Ri = −Qi or if Qi = Ri = 0. It is not likely Ri = 0 except in trivial cases. There appears to be no equivalent ignorable variables easily identified for the true non-holonomic case. In the holonomic case these equations reduce to Hamilton’s canonical equations so ignorable variables can be identified. It appears that it may be possible to constrain a path by external forces such that Ri = −Qi can be used to help define such a path, an optimization problem. In the case of energy conservation, we desire (writing E = E(pi , q j )): ∂E ∂E dE = p˙i + q˙ j , sum i ∈ I , j = 1, 2, ..., N dt ∂pi ∂q j ∂E ∂E = (Qi + Ri ) + (K jk + K jd Adk )Bki pi (38) ∂pi ∂q j
0=
This is a function of the momenta and generalized coordinates and any inputs in Qi . If it is identically zero then energy is conserved for the system as formulated. It is clear that this can be used to prescribe energy conserving forcing through Qi . This result is also an energy manifold partial differential equation over the 2N − M dimensional non-holonomic phase
space (q j , pi ) of the form:
gi (qk , pl )
∂E ∂E + h j (qk , pl ) =0 ∂pi ∂q j
(39)
sum on i ∈ I ; j = 1, 2, ..., N for l ∈ I ; k = 1, 2, ..., N. This may lead to insights needed to generate a Hamilton-Jacobi-like transformation theory extension for these “canonical” equations. 2.2
Non-Holonomic Lagrangian Method In this section we derive a canonical set of equations for non-holonomic systems via Hamilton’s principle. Following the work in [4] we can write the variation of the action integral as: 0 = δA Z t2 ∂L d ∂L δqi − Qi δqi dt = δqi − dt ∂q˙i ∂qi t1
(40)
where the generalized force Qi is given by equation 2. As before, we introduce an alternative set of variations via equation 22 giving:
0 = δA =
Z t2 d ∂L t1
∂q˙k
dt
−
∂L − Qk Kk j δu j dt ∂qk
0 = δA dt
(41)
q˙d = Kd j A jk Yki q˙i + Kd j A jk Zk + Jd
∂q˙k
d 0= dt
∂L ∂q˙k
(46)
which is developed via equations 8, 10 and 11, and from the fifth to sixth line we note that the d ∈ D index must always appear in the first location of the coupling tensor5 A so Ald Adi = 0. Equations 45 are linear equations in the independent q˙n and can be inverted as: q˙i = Bin pn
(47)
−
∂L − Qk (Kki + Kkd Adi ) δui dt ∂qk
where summation on k = 1, 2, ..., N, d ∈ D , and i ∈ I is implied. These variations δui are independent and arbitrary so the N − M equations of motion are:
sum on j ∈ I , and d, l ∈ D , for each i ∈ I . In the first and last line k, m = 1, 2, ..., N and use all n ∈ I . Note that from the second to third line we use:
(42)
Z t2 d ∂L t1
∂L (Kki + Kkd Adi ) ∂q˙ k ∂L ∂L ∂q˙d = + K ji + K jd Adi ∂q˙ j ∂q˙d ∂q˙ j ∂L ∂L = + (Kdl Alm Ym j ) K ji + K jd Adi (45) ∂q˙ j ∂q˙d ∂L ∂L (Kdl Alm ) (δmi + δmd Adi ) = K ji + K jd Adi + ∂q˙ j ∂q˙d ∂L ∂L K ji + K jd Adi + Kdl (Ali + Ald Adi ) = ∂q˙ j ∂q˙d ∂L ∂L = K ji + K jd Adi + Kdl Ali ∂q˙ j ∂q˙d = pi (qm , q˙n )
pi =
The variations are coupled via equation 10, so we chose N − M independent variations and M dependent variations and utilize equation 25 to write:
=
qk , they can be considered as Lagrange’s equations for nonholonomic systems. We continue from here and define generalized momenta as:
∂L − − Qk (Kki + Kkd Adi ) ∂qk
(43)
sum on n ∈ I for each i ∈ I . This is not the same coefficient tensor B shown is equation 31 but is also positive-definite. With this result, equation 46 becomes: q˙d = Kd j A jk Yki Bin pn + Kd j A jk Zk + Jd
sum on i, j, k = 1, 2, ..., N and n ∈ I for each d ∈ D . If we borrow the Kronecker delta we can combine the independent and dependent generalized coordinate derivative equations and write:
Summation on k = 1, 2, ..., N and on d ∈ D , for each i ∈ I . These equations are supplemented with the M nonholonomic constraint equations 10:
q˙l = δld (Kd j A jk Yki Bin pn + Kd j A jk Zk + Jd ) = (δld Kd j A jk Yki + δli )Bin pn +δld (Kd j A jk Zk + Jd )
Nk = ak j (Y jl q˙l + Z j ) = 0
(44) where l = 1, 2, 3, ..., N.
sum on j, l = 1, 2, ..., N for each k = 1, 2, ..., M, where equation 8 has been used for the speeds. These (equations 43 and 44) are the equations needed to solve for N unknown
(48)
5 Not
a true tensor, it is a coefficient matrix.
(49)
Using equations 45, equations 43 becomes: p˙i = Qi + Ri
(50)
with:
ergy is conserved for the system as formulated. The discussion about energy conservation in section 2.1.1 also applies here.
3 Qi = Qk (Kki + Kkd Adi ) +
∂L (Kki + Kkd Adi ) ∂qk
(51)
where Qk is given by equation 2 for any polygenic forces and moments, sum on k = 1, 2, ..., N for each i ∈ I . Also where: ∂L d (Kki + Kkd Adi ) ∂q˙k dt ∂L ∂Kki ∂Kki ∂Kkd ∂Kkd = q˙ j + q˙d + q˙ j + q˙l Adi ∂q˙k ∂q j ∂qd ∂q j ∂ql ∂Adi ∂Adi q˙l q˙ j + (52) +Kkd ∂q j ∂ql = Ri (qk , pm ), k = 1, 2, ..., N, m ∈ I
Ri =
In the second line sum on k = 1, 2, ..., N, and j ∈ I and d, l ∈ D . The q˙ j are replaced by equations 47 and the q˙l are replaced by equations 48. If Qk is a function of q˙k they can be eliminated via equations 47 and 48 as appropriate. Equations 49-52 comprise a set of 2N − M ordinary differential equations of motion for non-holonomic systems. They are canonical in the sense they are fundamental and formulaic but lose the symmetry of the Hamilton-Jacobi canonical equations yet gain that they are non-DAE. These equations do not lend as much physical insight into how the generalized momentum relate to the vector momenta of equations 28, but follow the general and common formulation techniques of Lagrange’s equation.
Examples We present three sets of examples of the comparative use of the equations of motion presented above. The first problem is a particle restricted to move on a differentially defined curve, the second is the problem of a disk or hoop rolling without slip on a plane surface, and the third problem, a planar mechanism, serves as a multiple body example, and again shows how the mapping from the momentum space back to the full configuration space (defined with transcendental coupling of coordinates) is inherent in the methodology. 3.1
Simple Particle System A particle of mass m moving in the x-y-plane under gravity is subject to a non-holonomic constraint6 :
N = y˙ − y2 x˙ = 0 The velocity of the particle is: ~v = x˙nˆ 1 + y˙nˆ 2
Comments on Momentum and Energy Conservation The comments regarding momenta conservation above in section 2.1.1 apply here as well. The energy conservation expression is of slightly different form (again we write E = E(pi , ql ), i ∈ I ; l = 1, 2, 3, ..., N): dE dt ∂E ∂E = p˙i + q˙l , sum i ∈ I ; l = 1, 2, ..., N ∂pi ∂ql ∂E = (Qi + Ri ) ∂pi ∂E + (δld Kd j A jk Yki + δli )Bin pn ∂ql +δld (Kd j A jk Zk + Jd )
0=
(55)
where the x-unit vector is nˆ 1 and the y-unit vector is nˆ 2 . We assign the generalized speeds as: u1 = x˙ and
u2 = y. ˙
(56)
So equation 9 provides that:
K11 = K22 = 1 and K12 = K21 = 0. 2.2.1
(54)
(57)
The coefficients from equation 3 are: a11 = −y2
and a12 = 1
(58)
We arbitrarily choose u2 as independent, so the constraint coupling term from equations 11, 54 and 56 is: A12 = −
1 a12 = 2 a11 y
(59)
(53)
Note that we can also choose x˙ as the dependent speed and avoid the apparent singularity when y → 0 for this problem.
This is a function of the momenta and generalized coordinates and any inputs in Qi . If it is identically zero then en-
6 This is a contrived constraint to demonstrate the technique on a simple planar problem. This constraint is in fact integrable, y = −1/x + c so this problem is actually holonomic. Possibly the simplest truly non-holonomic particle system is a particle moving in space with N = y˙ + zx˙ = 0 as the motion constraint. This three dimensional non-holonomic particle problem is studied thoroughly elsewhere [18, 20].
Hamilton Canonical Formulation 0
2.0
-10
1.5 y[t]
-20 x[t]
As the hyperbolic solutions show, we do not have a singularity for this choice of dependent speeds. The kinetic energy is:
-30 -40
1.0 0.5
-50
1 T = m(x˙2 + y˙2 ) 2
-60
(60)
0.0 0
2
4
6
8
10
0
2
-1 p 2 [t]
(61)
p 1 [t]
-2
V = mgy
-3 -4 -5 -6 0
For the projective equations, the active force is:
4
λ[t]
(62)
The Lagrangian is:
6
8
10
0
20
8.× 10-7
15
6.× 10-7
10
0
2
4
(63)
(64)
6
8
∂L = mx˙ and ∂x˙
p2 =
∂L = my˙ ∂y˙
(65)
p1 m
y˙2 =
p2 m
(66)
then and
p21 + p22 + 2m2 gy 2m
0
2
Q02 = λa12 = λ
(68)
8
10
2.0
y[t]
1.5 1.0 0.5 2
4
6
8
10
0.0 -70
-60
-50
Fig. 1.
-40
-30
-20
-10
0
x[t]
Response Plots Hamilton’s Canonical Equations 69
These DAE are solved7 with m = 1 and g = 10 with initial quiescent speeds/momenta and at space point (2,2) for ten seconds. The responses are plotted in figure 1. Shown are (from top left to bottom right): x(t), y(t), p1 (t), p2 (t), λ(t), N (t), E(t) and y verses x.
u2 nˆ 1 + u2 nˆ 2 y2
(70)
From equation 14, with ~L = m~v, we have: ! ~L ∂~v d · ~F − 0= ∂u2 dt 1 = nˆ 2 + 2 nˆ 1 · y u˙2 2u2 −mg nˆ 2 + m u˙2 nˆ 2 + 2 − 32 y y
The equations of motion are from equations 7: ∂H + Q01 = −y2 λ ∂x ∂H p˙2 = − + Q02 = −mg + λ ∂y ∂H p1 x˙ = = ∂p1 m ∂H p2 y˙ = = ∂p2 m p y2 or p N = y˙ − y2 x˙ = 2 − 1 = 0 m m
6
(67)
The non-holonomic generalized forces are (see equations 4): and
4 t (s)
~v =
Q01 = λa11 = −λy2
10
3.1.2 Projective Dynamical Equations Here we write the velocity as a function of only the independent generalized speed, so:
Using this, the Hamiltonian is: H = p1 x˙ + p2 y˙ − L =
8
2.× 10-7
-2.× 10-7
10
20 20 20 20 20 20 20 0
3.1.1 Hamilton’s Canonical Equations Define the generalized momentum via equations 5 as:
6
4.× 10-7
t (s)
x˙1 =
4
t (s)
E[t]
1 E = T +V = m(x˙2 + y˙2 ) + mgy 2
2
0
0
The total energy is:
10
t (s)
5
1 L = T −V = m(x˙2 + y˙2 ) − mgy 2
8
0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0
t (s)
Nh[t]
~F = −mgnˆ 2
2
6 t (s)
0
and the potential energy is:
p1 =
4
t (s)
p˙1 = −
(69) 7 The reader is forewarned that the author did not attempt to force the numerical integrators utilized in what follows to conserve energy. It is instructive to see which formulation provides the equation structure that allows the default algorithm chosen by the integration logic to best conserve energy. The integrators do provide options that allow the conserved energy to be maintained; they were not utilized in what follows.
Projective Formulation 0
2.0
-10
1.5
3.1.3 Projective Canonical Equations In this section we use the first of the derived canonical formulations for the stated particle problem. We use the generalized coordinates and generalized speeds for the formulation. The velocity is:
y[t]
x[t]
-20 -30 -40
1.0 0.5
-50 -60
0.0 0
2
4
6
8
10
0
2
4
0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 0
2
4
6
8
10
0.00003 0.00002 0.00001 0 -0.00001 -0.00002 -0.00003 -0.00004
8
10
~v = u1 nˆ 1 + u2 nˆ 2
0
2
4
6
8
10
∂~v ∂~v + A12 ∂u2 ∂u1 1 = m(u1 nˆ 1 + u2 nˆ 2 ) · nˆ 2 + nˆ 1 2 y u1 1 = m u2 + 2 = m 1 + 4 u2 y y
p2 = m~v ·
t (s) 2.0
20.0000
1.5 y[t]
19.9998 19.9996 19.9994
1.0 0.5
19.9992 0
2
4
6
8
10
0.0 -70
-60
-50
t (s)
Fig. 2.
=
-40
-30
-20
-10
0
(74)
x[t]
where we use u1 = A12 u2 = u2 /y2 from the non-holonomic constraint 59 in the last line. Solving for the speed u2 we get:
Response Plots Projective Equations 71 and 72.
(m(−2u22 + y(u˙2 + y4 (g + u˙2 )))) y
u2 =
(71)
= −2u22 + y(u˙2 + y4 (g + u˙2 ))
u2 y2
and y˙ = u2 .
p2 y4 m(1 + y4 )
(75)
The generalized force from equations 33 is:
as the equation of motion, and the kinematic equations of motion are: x˙ =
(73)
The momenta is from equation 28:
t (s)
E[t]
6 t (s)
Nh[t]
u 2 [t]
t (s)
∂~v ∂~v + A12 ∂u2 ∂u1 1 = −mg = −mgnˆ 2 · nˆ 2 + nˆ 1 2 y
Q2 = ~F ·
(72)
Solving these equations under the same parameters, time and initial conditions gives the results shown in figure 2. If the time of integration is normalized over the time taken for equations 69, these equations are 95% faster, using default solvers chosen by the algorithm in Mathematica (version 10.3.1 on OSX) NDSolve routine [21]. Plotted are (from top left to bottom right): x(t), y(t), u2 (t), N (t), E(t) and y verses x. Comparing figures 1 and 2 we see that p2 in the former is directly proportional to u2 8 in the latter, and surprisingly, the energy (20 units) and constraint are maintained better in the former than the latter. The integrator used is slightly dissipative in the latter case. No attempt was made to force the integrators to track energy conservation. There is virtually no difference in the planar trajectory and displacement time responses. The constraint appears to be better satisfied by a factor of 100 initially in the former integration, and both approaches converge to zero as desired, and constraint divergence from zero is no bigger than ±4.0 × 10−5 .
(76)
The remaining term from equations 35 is: ∂~v ∂~v + A12 ∂u2 ∂u1 1 d = m(u1 nˆ 1 + u2 nˆ 2 ) · nˆ 2 + nˆ 1 2 dt y
P2 = ~L ·
d dt
(77)
after writing this equation in terms of ps and qs only, then
R2 = −
2p22 y3 m(1 + y4 )2
(78)
From equations 36 and 37 the equations of motion are:
p˙2 = Q2 + R2 = −mg −
2p22 y3 m(1 + y4 )2
(79)
and 8 Because
case.
of the simple selection of the coupling coefficients Ki j in this
x˙ = A12 u2 =
p2 y2 m(1 + y4 )
and
y˙ =
p2 y4 m(1 + y4 )
(80)
Projective Canonical Formulation 0
2.0
-10
1.5
which are two of the needed equations of motion. From equation 51:
y[t]
x[t]
-20 -30 -40
1.0 0.5
∂L ∂L Q2 = 0 + (K12 + K11 A12 ) + (K22 + K21 A12 ) ∂x ∂y 1 ∂L 1 ∂L 0 + (1) 2 + 1 + (0) 2 (84) = ∂x y ∂y y 1 1 + (−mg) 1 + (0) 2 = (0) 0 + (1) 2 y y = −mg
-50 -60
0.0 0
2
4
6
8
10
0
2
4
t (s)
Nh[t]
p 2 [t]
-5000 -10 000 -15 000 -20 000 2
4
6
8
10
0
2
4
6
8
10
t (s)
From equation 52:
2.0
20.0000
1.5 y[t]
19.9998 E[t]
10
0.00002 0.000015 0.00001 5.× 10-6 0 -5.× 10-6 -0.00001 -0.000015
t (s)
19.9996
1.0
0
2
4
6
8
10
0.0 -70
t (s)
∂L d ∂L d (K12 + K11 A12 ) + (K22 + K21 A12 ) (85) ∂x˙ dt ∂y˙ dt ∂L d 1 ∂L d 1 = 0 + (1) 2 + 1 + (0) 2 ∂x˙ dt y ∂y˙ dt y 1 d 1 d 0 + (1) 2 + (my) ˙ 1 + (0) 2 = (mx) ˙ dt y dt y d 1 d = (my/y ˙ 2) + (my) ˙ (1) dt y2 dt
P2 =
0.5
19.9994
Fig. 3.
8
t (s)
0
0
6
-60
-50
-40
-30
-20
-10
0
x[t]
Response Plots Projective Momentum Equations 79 and 80.
Solving these equations under the same parameters and initial conditions gives the results shown in figure 3. Indicated are (from top left to bottom right): x(t), y(t), p2 (t), N (t), E(t) and y verses x. The speed up is 94% relative to the Hamilton canonical formulation. We notice here the integration is slightly dissipative and the constraint is satisfied within ±20.0 × 10−6 . This momentum p2 is much larger than the momenta of the Hamiltonian formulation. The spatial coordinates x and y track as before. 3.1.4 Non-Holonomic Lagrangian Canonical Equations Here we solve the same example but with the last algorithm presented above. The generalized momentum from equation 45 is: ∂L ∂L (K22 + K21 A12 ) + K11 A12 ∂y˙ ∂x˙ 1 1 = my˙ 1 + (0) 2 + mx(1) ˙ y y2 y˙ 1 m(1 + y4 )y˙ = my˙ + m 2 = y y2 y4
(82)
then using the x˙ above:
x˙ =
(86)
where we have used equation 82 in the last line. The last equation of motion is from equation 50:
p˙2 = Q2 + R2 = −mg −
2p22 y3 m(1 + y4 )2
(87)
3.1.5 Examples Summary We have demonstrated two standard techniques and two new “canonical” techniques to formulate equations of motion. The simple example allows exposition of each term in the equations so that comparisons can be made. We provided rudimentary numerical comparisons as well. For this problem the new canonical techniques were not the fastest, but they were 94% faster than Hamilton’s canonical formulation. We will test computation speed again with the complex motion example to follow. 3.2
p2 y2 m(1 + y4 )
2p22 y3 m(1 + y4 )2
(81)
where we have used x˙ = y/y ˙ 2 in the third line. Inverting this gives: p2 y4 m(1 + y4 )
R2 = −
These equations are the same as the projective equations so no numerical comparison is required.
p2 =
y˙ =
after writing this equation in terms of ps and qs only, then
(83)
Rolling Disk We present this example in sparse detail due to the volume of equations but in enough detail to highlight the merits of the new canonical techniques. We use Mathematica and
a set of vector tools [22] that allows Gibbsian vector-dyadic notation to be used. We define the x-y-plane as the plane the disk is rolling on. Unit vectors nˆ i , i = 1, 2, 3 are in the x, y, z directions respectively for the Newtonian frame. The contact point is located on the plane with q4 , q5 in the nˆ 1 , nˆ 2 , directions respectively. The disk heading, tangent to the planar path, is pointed along 1-direction of an intermediate frame A that is rotated with and Euler 3-rotation (q1 ) from the Newtonian frame. Another intermediate frame B tilts with the disk under an Euler 1-rotation (q2 ), and lastly the body fixed frame C Euler 2-rotates (q3 ) from the tilted frame and denotes the rolling of the disk. Refer to figure 4 for visual reference. The disk has radius R, its mass mr , there is allowed an eccentric mass ms located (h, e, r) in frame C from the geometric center of the disk.
and cos(q3 ) 0 − sin(q3 ) bˆ 1 cˆ1 bˆ 2 cˆ2 = 0 1 0 ˆ cˆ3 sin(q3 ) 0 cos(q3 ) b3
(91)
The body-fixed angular velocity is: ~ω = u1 cˆ1 + u2 cˆ2 + u3 cˆ3
(92)
where the generalized speeds or quasi-coordinates are given by: u1 = q˙2 cos(q3 ) − q˙1 cos(q2 ) sin(q3 ) u2 = sin(q2 )q˙1 + q˙3 u3 = cos(q2 ) cos(q3 )q˙1 + sin(q3 )q˙2
(93)
u4 = q˙4 u5 = q˙5 So from equation 8 we identify:
Y11 = − cos(q2 ) sin(q3 ), Y12 = cos(q3 ), Y13 = 0, Y14 = 0, Y15 = 0 Y21 = sin(q2 ), Y22 = 0, Y23 = 1, Y24 = 0, Y25 = 0 Y31 = cos(q2 ) cos(q3 ), Y32 = sin(q3 ), Y33 = 0, Y34 = 0, Y35 = 0 (94) Y41 = 0, Y42 = 0, Y43 = 0, Y44 = 1, Y45 = 0 Y51 = 0, Y52 = 0, Y53 = 0, Y54 = 0, Y55 = 1 The inverse relationships are: Fig. 4.
Base, heading, and tilt frames of disk/ring.
q˙1 = sec(q2 )(− sin(q3 )u1 + cos(q3 )u3 ) q˙2 = cos(q3 )u1 + sin(q3 )u3 The angular velocity of the disk is: ~ω = q˙1 aˆ3 + q˙2 bˆ 1 + q˙3 cˆ2
q˙3 = sin(q3 ) tan(q2 )u1 + u2 − cos(q3 ) tan(q2 )u3 (95) q˙4 = u4 (88)
The unit vectors transform as:
q˙5 = u5 We make note of the singularity near q2 = π/2. From equations 9 we identify:
cos(q1 ) sin(q1 ) 0 nˆ 1 aˆ1 aˆ2 = − sin(q1 ) cos(q1 ) 0 nˆ 2 aˆ3 0 0 1 nˆ 3
(89)
1 0 0 bˆ 1 aˆ1 bˆ 2 = 0 cos(q2 ) sin(q2 ) aˆ2 ˆ 0 − sin(q2 ) cos(q2 ) aˆ3 b3
(90)
and
K11 = − sec(q2 ) sin(q3 ), K12 = 0, K13 = sec(q2 ) cos(q3 ), K14 = 0, K15 = 0 K21 = cos(q3 ), K22 = 0, K23 = sin(q3 ), K24 = 0, K25 = 0 K31 = sin(q3 ) tan(q2 ), K32 = 1, K33 = − cos(q3 ) tan(q2 ), K34 = 0, K35 = 0 (96) K41 = 0, K42 = 0, K43 = 0, K44 = 1, K45 = 0 K51 = 0, K52 = 0, K53 = 0, K54 = 0, K55 = 1
The non-holonomic constraints for rolling without slip are:
N1 = q˙4 − Rq˙3 cos(q1 ) = u4 − R cos(q1 )(sin(q3 ) tan(q2 )u1 + u2 − cos(q3 ) tan(q2 )u3 ) = 0
(97)
N2 = q˙5 − Rq˙3 sin(q1 ) = u5 − R sin(q1 )(sin(q3 ) tan(q2 )u1 + u2
Choosing u4 and u5 as dependent speeds, from equations 11 we identify: A41 = R cos(q1 ) sin(q3 ) tan(q2 ), A42 = R cos(q1 ), (98)
A51 = R sin(q1 ) sin(q3 ) tan(q2 ), A52 = R sin(q1 ), A53 = −R sin(q1 ) cos(q3 ) tan(q2 ) For this problem N = 5, M = 2, so I = {1, 2, 3} and D = {4, 5}. The number of bodies is NB = 1. The position vector to the center of mass is: ~r = q4 nˆ 1 + q5 nˆ 2 + Rbˆ 3 + hcˆ1 + ecˆ2 + rcˆ3
~ = T f u21 sgn (u1 ) cˆ1 + u22 sgn (u2 ) cˆ2 + u23 sgn(u3 )cˆ3 M (104) The translational and angular momenta for the projective method are: ~L = m~v
− cos(q3 ) tan(q2 )u3 ) = 0
A43 = −R cos(q1 ) cos(q3 ) tan(q2 )
We added an air drag body torque:
(99)
The velocity of the center of mass is: ~v = (Ru3 sin (q3 ) + Ru1 cos (q3 )) aˆ1 × bˆ 3 + (Ru3 cos (q3 ) sec (q2 ) − Ru1 sin (q3 ) sec (q2 )) aˆ3 × bˆ 3 + (eu1 − hu2 ) cˆ3 + (ru2 − eu3 ) cˆ1 + (hu3 − ru1 ) cˆ2 + (Ru2 cos (q1 ) − Ru3 cos (q1 ) cos (q3 ) tan (q2 ) +Ru1 sin (q3 ) cos (q1 ) tan (q2 )) nˆ 1
~ =~~I · ~ω and H
(105)
3.2.1 Projective Formulation In this simulation the equations of motion are generated with Kane’s form of the Gibbs-Appell equations described in section 1.2. The disk initial speeds are zero except q˙3 [0] = 20 rad/s. The angles are zero except a slight tilt q2 [0] = 0.15 rad. Gravity is taken as 981cm/s2 , the disk mass is mr = 30 g, the radius R = 1.5 cm, and the eccentric mass ms = 1g located at r = 0.95R in the plane of the disk (e = h = 0). The central mass moment of inertia was used for the disk shifted to the center of mass and corrected with the inertia of the eccentric mass. No drag was prescribed. The simulation was run for 10 seconds, and took a normalized 1 unit of time for the default integrator of NDSolve in Mathematica. Shown in figure 5 are u1 (t), u2 (t), u3 (t), q1 (t), q2 (t), q3 (t), q4 (t), q5 (t), q5 (t) vs q4 (t), and E(t), from top left to bottom right. Of significance is that the integrator does allow deviation of the total energy, about 13% percent of the total. These deviations occur when the disk is approaching horizontal as it rolls and dips; this dipping motion is rapid compared to the overall rolling motion so it is suspected that the integrator did not step with sufficient fidelity. No attempt was made to choose integrator options that could mitigate these deviations in energy. Also the disk wandered away from the starting origin as can be seen at the lower left plot of figure 5. Figure 4 is a snapshot of the typical motion of the disk.
+ (Ru2 sin (q1 ) + Ru1 sin (q1 ) sin (q3 ) tan (q2 ) −Ru3 sin (q1 ) cos (q3 ) tan (q2 )) nˆ 2
(100)
(102)
3.2.2 Projective Canonical Formulation-u’s The parameters and initial conditions are the same in this simulation but the equations of motion were generated as presented in section 2.1. We also define the generalized speeds as defined above in equation 93. The simulation was run for 10 seconds, and took, relative to the baseline time above, 16% less time for the default integrator of NDSolve in Mathematica. Shown in figure 6 are p1 (t), p2 (t), p3 (t), q1 (t), q2 (t), q3 (t), q4 (t), q5 (t), q5 (t) vs q4 (t), and E(t), from top left to bottom right. Of significance is that the integrator does again allow deviation of the total energy, but a smaller percentage (7 %) of the total than the case in section 3.2.1, and show an increase in energy for the numerical deviations. The disk also does not wander away from the origin.
(103)
3.2.3 Projective Canonical Formulation-q’s ˙ Everything is the same in this simulation but the equations of motion were generated as presented in section 2.1.
The kinetic energy is: 1 1 T = m~v ·~v + ~ω ·~~I · ~ω 2 2
(101)
where ~~I is in the inertia dyad of the disk shifted to the offset center of mass. For the Lagrangian formulation, the velocity and angular velocities are written in terms of the q’s ˙ via equations 93. The potential energy is: V = mg~r · nˆ 3 For the projective formulation, the active force is: ~F = −mgnˆ 3
u-Projective Canonical Formulation
Projective Formulation 15 24
0 -5
22
0
2
4
6
8
10
0
2
4
6
t (s)
8
0
10
6
8
10
0
0
4
6
8
0
10
2
4
6
8
0
10
0.8
1.0
250
0.8
200
q 2 [t]
1.0
300
q 3 [t]
350
1.2
150 100
0.4
4
6
8
10
6
8
-80
10
0
2
4
6
t (s)
8
0.4
2
4
6
8
10
4
6
8
10
0
-2
40
0 -2
4
6
8
10
-8 -10 0
2
4
6
8
10
0
72 000
-2
78 000
q 5 [t]
79 000
60
0
20
40 q 4 [t]
60
80
8
10
80 000
0 -4
68 000
-6
66 000
-8
64 000 2
4
6
8
10
t (s)
Fig. 5. Response Plots Projective Momentum Equations 15 and 16 for Rolling Disk.
But different from above, we define ui = q˙i , i = 1, 2, 3, 4, 5. Then we use the angular velocity as shown in equation 88. The simulation was run for 10 seconds, and took, relative to the baseline time above, 67% less time for the default integrator of NDSolve in Mathematica. Shown in figure 7 are p1 (t), p2 (t), p3 (t), q1 (t), q2 (t), q3 (t), q4 (t), q5 (t), q5 (t) vs q4 (t), and E(t), from top left to bottom right. This time the integrator numerical errors caused decreased energy deviation, but again a smaller percentage (3 %) of the baseline energy. The disk does not wander from the origin. Note also that the momenta pi , i = 1, 2, 3 are unique for this choice of generalized speeds as compared to the choice in the case in section 3.2.2. 3.2.4 Lagrangian Canonical Formulation Everything is the same in this simulation but the equations of motion were generated as presented in section 2.2.
77 000 76 000 75 000
-10 0
4 t (s)
2
70 000
2
t (s)
74 000
0
6
-6
-6
80
20
10
-4
-4
t (s)
E[t]
q 5 [t]
q 5 [t]
0
60
2
4
2
2
0
2
t (s)
4
t (s)
40
8
0 2
E[t]
0
6
100
80
0
0
10
150
t (s)
20
20
8
50 0
10
q 4 [t]
q 5 [t]
40
4
200
0.6
6
60
2
250
t (s)
80
0
t (s)
0.0
0 2
4
0.2
50
0.2
6
-40
t (s)
1.4
0
2
t (s)
t (s)
0.6
10
-60
-1500
q 3 [t]
2
8
-20
500
-1000 -150 0
6
-500
-100
-15
4
0
q 1 [t]
p 3 [t]
0
2
t (s)
1000 -50 q 1 [t]
u 3 [t]
4
1500
-10
q 2 [t]
2
2000
0
-5
3000
t (s)
10 5
3100
2900
t (s)
15
q 4 [t]
-500 -1500
20
-15
0 -1000
21
-10
3200
500
23
p 1 [t]
u 2 [t]
5 u 1 [t]
1000 p 2 [t]
10
3300
1500
25
-6 -4 -2 0 q 4 [t]
2
4
6
0
2
4
6
8
10
t (s)
Fig. 6. Response Plots u-Projective Momentum Equations 36 and 37 for Rolling Disk, with the u’s defined in equation 93.
We use the generalized speeds shown in equations 93 to implement the constraints as depicted in the methodology. The simulation was run for 10 seconds, and took, relative the baseline time above, 81% less time for the default integrator of NDSolve in Mathematica. Shown in figure 8 are p1 (t), p2 (t), p3 (t), q1 (t), q2 (t), q3 (t), q4 (t), q5 (t), q5 (t) vs q4 (t), and E(t), from top left to bottom right. In this formulation the energy deviations is very small, so the default integrator has less trouble with the equations as formulated. Note momenta pi , i = 1, 2, 3 are the same generalized momenta as in the case of section 3.2.2. It is thought that the integrator speed differential and accuracy are due to the fact that the singularity at the inversion in equations 96 is not present in this formulation. The path of the disk is also different than section 3.2.2, and is probably due to the fact this default integration tracked conserved energy much better. But these issues are not really investigated here, and is reason for further study when choosing/developing appropri-
q'-Projective Canonical Formulation
Lagrangian Canonical Formulation 3200
1800
1200
p 2 [t]
p 1 [t]
1400 1000
3150
500
3100
p 1 [t]
500 0
p 2 [t]
1600
1000
0 -500
800 -500
600
2900
4
6
8
0
10
2
4
4
8
2850
10
0
0
1000
-10
-40 -60
0
4
6
8
10
0
2
4
6
t (s)
8
10
0.4
250
0.8
200
0.6
150 100
4
4
6
8
10
4
6
8
2
4
6
q 4 [t]
q 5 [t]
8
-8 0
2
4
6
8
10
2
4
10
6
8
10
t (s) 0 -4
5
0 2
4
6
8
-8 -12
-5 0
-6 -10
2
4
6
10
8
10
0
2
4
t (s)
t (s)
t (s) 75 198.5
0 -2
0
74 500
-4
-2 -4
74 000
-6
-6
-10
73 000
-12
-8
72 500 2
4
6
q 4 [t]
75 197.0 75 196.5 -5
0
2
4
6
8
75 197.5
-8
73 500
-10
75 198.0 E[t]
75 000 q 5 [t]
2
E[t]
q 5 [t]
8
100
0
0
t (s)
-6 -4 -2 0
6
150
10
-6 -10
10
-2
-4
-6
8
t (s)
10
-2
-4
4
t (s)
0
-2
2
50
0
10
2
0
0
250
t (s)
2
10
0 2
t (s)
4
8
0.2
0
6
6
0.4
0 2
6
-40
200
50
0.2
10
-30
t (s)
q 2 [t]
q 3 [t]
0.6
0
2
t (s)
0.8
8
-60 0
q 3 [t]
2
6
-50
-1000
q 5 [t]
0
4
-20
500
-500
2900
2
t (s)
1500
p 3 [t]
q 1 [t]
3000
6 t (s)
-20
3100 p 3 [t]
10
0
3200
q 2 [t]
8
2
t (s)
t (s)
q 4 [t]
6
0
q 1 [t]
2
3000 2950
-1000
400 0
3050
0
5
10
10
0
2
4
6
8
10
t (s)
q 4 [t]
t (s)
Fig. 7. Response Plots q˙-Projective Momentum Equations 36 and 37 for Rolling Disk, with ui = q˙i .
Fig. 8. Response Plots Lagrange Momentum Equations 47-52 for Rolling Disk.
n2
ate numerical integrators.
q3
3.3
a1
q2
b1
q1
a2
Planar Slider Crank In this example we present the modeling of a planar quick return mechanism. This example demonstrates the canonical technique on multiple body systems. The example also re-emphasizes the differential mapping of the N − M dimensional momentum hyperspace back into the N dimensional configuration space. The schematized quick return mechanism and generalized coordinates are depicted in figure 9. We assume the crank (mass ma ) is driven by a torque motor Tm at ao and the follower (mass mb ) has a load torque at Td at bo . The frames of reference and the geometric parameters are defined in the figure. Frictional loading in the pin slot in the follower is ignored for this demonstrative example.
H
B
A n1
ao
bo
b2
Fig. 9.
Quick Return Mechanism
The vector loop constraint for the device is: 0 = −H aˆ1 + Anˆ 1 + q3 (t)bˆ 1
(106)
4Ai33 q23 + B2 H 2 mb cos2 (q1 − q2 )
Projecting this along the nˆ 1 and nˆ 2 directions gives, respectively, two transcendental holonomic constraints:
+4Bi33 H 2 cos2 (q1 − q2 ) + H 2 ma q23
+2p21 q3 B2 H 3 mb sin(3q1 − 3q2 ) + B2 H 3 mb sin(q1 − q2 ) +4Bi33 H 3 sin(3q1 − 3q2 ) + 4Bi33 H 3 sin(q1 − q2 ) /
c1 = A − H cos(q1 ) + q3 cos(q2 ) = 0 c2 = q3 sin(q2 ) − H sin(q1 ) = 0
(107)
4Ai33 q23 + B2 H 2 mb cos2 (q1 − q2 ) +4Bi33 H 2 cos2 (q1 − q2 ) + H 2 ma q23 −2BgHmb cos(q1 − 2q2 ) − 2BgHmb cos(q1 ) + 8q3
We chose to keep the configuration space over determind, but do choose the crank angle q1 to be the independent holonomic variable for this one DOF problem. For the canonical methodology we elevate the constraints to the speed level as:
N1 = H sin(q1 )q˙1 − q3 sin(q2 )q˙2 + cos(q2 )q˙3 = 0 (108) N2 = −H cos(q1 )q˙1 + q3 cos(q2 )q˙2 + sin(q2 )q˙3 = 0 We will choose the generalized speeds for the defintion of the constraint coupling terms simply as ui = q˙i , i = 1, 2, 3 so that Ki j = δi j . The speed constraints, equations 108 can be solved for the dependent speeds as: Hu1 cos(q1 − q2 ) q3 u3 = −Hu1 sin(q1 − q2 )
q˙1 = 4p1 q23 / 4Ai33 q23 + B2 H 2 mb cos2 (q1 − q2 ) +4Bi33 H 2 cos2 (q1 − q2 ) + H 2 ma q23 (114) The second mapping to the configuration equation is from equation 49: q˙2 = 4H p1 q3 cos(q1 − q2 )/ +4Bi33 H 2 cos2 (q1 − q2 ) + H 2 ma q23
(111)
We define the generalized momentum via equation 45 as:
H cos(q1 − q2 )
+
B2 Hmb q˙1 cos(q1 −q2 ) 4q2
(112)
1 −q2 ) + Bi33 H q˙1 cos(q q2
q2
The inverse relationship q˙1 = f (p1 , q1 , q2 , q3 ) is easily found and shown in equation 114 below. From equation 50, the generalized momentum differential equation becomes: 1 HTd cos(q1 − q2 ) p˙1 = − gHma cos(q1 ) + + Tm 2 q3
(113) 2p21 q23 −B2 H 2 mb sin(2q1−2q2 ) − 4Bi33 H 2 sin(2q1−2q2 ) /
(115)
The final configuration mapping from equation 49 is: q˙3 = −4H p1 q23 sin(q1 − q2 )/ 4Ai33 q23 + B2 H 2 mb cos2 (q1 − q2 )
(110)
where Ai33 and Bi33 are centroidal mass moments of inertia. The potential energy is:
1 p1 = Ai33 q˙1 + H 2 ma q˙1 4
4Ai33 q23 + B2 H 2 mb cos2 (q1 − q2 )
(109)
The constraint coupling terms become A21 = H cos(q1 − q2 )/q3 and A31 = −H sin(q1 − q2 ). The kinetic energy is:
1 1 V = mb gB sin(q2 ) + ma gH sin(q1 ) 2 2
2
The first configuration differential equation is from the inversion of the definition of the generalized momentum in equation 112 as described in equation 47:
u2 =
1 1 1 1 T = Ai33 q˙21 + H 2 ma q˙21 + Bi33 q˙22 + B2 mb q˙22 2 8 2 8
2
+4Bi33 H 2 cos2 (q1 − q2 ) + H 2 ma q23
(116)
These are the equations of motion for the device. They are the canonical set of equations for this system. Conservation of energy can be confirmed as indicated in equation 38 after differentiation of the total energy, T +V in equations 110 and 111, and using the equations of motion for the derivatives of the momenta and coordinates with the driving and load torque set to zero. In the normal formulation of a model such as this, it may not be as straight forward to see if energy is conserved, due to the differential algebraic nature of those common formulation schemes. A simulation was run with the parameters (mks units): A = 1; B = A + 1.5H; H = A/3; ma = 1/2; mb = 1;g = 9.81; Tm = Td = 0. The initial coordinates are as depicted in figure 9 π: q2 = 2.80729; q3 = 1.0034. The undamped 9: q1 = 20 response is shown in figure 10. Shown are the momentum p1 , the configuration coordinates, the constraints c1 and c2 from equations 107, and the total energy. The default integrator in NDSolve was used and the 10 second simulation took 0.056 seconds.
4
Summary and Conclusion We have presented two canonical formulations of the equations of motion for non-holonomic rigid-body mechan-
Quick Return Lagrangian Canonical Formulation 7
1.0
6 q 1 [t]
p 1 [t]
0.5 0.0 -0.5
5 4 3 2
-1.0 0
2
4
6
8
10
0
2
4
3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8
8
10
1.3 1.2 1.1 1.0 0.9 0.8 0.7 0
2
4
6
8
10
0
2
4
t (s)
6
8
10
t (s)
2.× 10-6
3.22157
1.5 × 10-6
3.22156
1.× 10-6
3.22155
E[t]
c1 [t], c2 [t]
6 t (s)
q 3 [t]
q 2 [t]
t (s)
5.× 10-7
3.22154 3.22153
0
3.22152 0
2
4
6 t (s)
Fig. 10.
8
10
0
2
4
6
8
10
t (s)
Quick Return Mechanism Undamped Free Reesponse
ical systems. We developed the methodologies from two perspectives developed from classical variational mechanics. We have compared the new methods with Hamilton’s Canonical equations for a non-holonomic system as well as Kane’s form of the Gibbs-Appell equations. We show how the new canonical equations are superior to the current state-of-theart for modeling these systems. It is evident that mathematically rigorous studies of appropriate quadrature techniques are required and also a parallel to the Hamilton-Jacobi differential transformations as solutions to these equations may be available with further insight.
Acknowledgement The author would like to extend gratitude to the Institute for Computational Engineering and Sciences at The University of Texas at Austin and their J.T. Oden Faculty Fellowship, and the Texas Tech University Faculty Development leave program for the funds to support these research efforts.
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[4] Lanczos, C., 1970. The Variational Principles of Mechanics, 4 ed. The University of Toronto Press, Toronto. Unabridged Dover republication, 1986. [5] Gibbs, J. W., 1961. The Scientific Papers of J. Willard Gibbs, Volume II: Dynamics, Etc. Dover, New York. [6] Kane, T. R., and Levinson, D. A., 1985. Dynamics Theory and Applications. McGraw-Hill, New York. [7] Desloge, E. A., 1988. “Efficacy of the Gibbs-Appell method”. Journal of Guidance, Control, and Dynamics, 12(1), January-February, pp. 114–115. [8] Neimark, J. I., and Fufaev, N. A., 1972. Dynamics of Non-Holonomic Systems. American Mathematical Society, Providence, Rhode Island. Translated from the Russian by J.R. Barbour. [9] Barhorst, A. A., and Everett, L. J., 1995. “Modeling hybrid parameter multiple body systems: A different approach”. The International Journal of Nonlinear Mechanics, 30(1), pp. 1–21. [10] Barhorst, A. A., and Everett, L. J., 1995. “Contact/impact in hybrid parameter multiple body systems”. Journal of Dynamic Systems, Measurement, and Control, 117(4), pp. 559–569. [11] Barhorst, A. A., 1998. “On modeling variable structure dynamics of hybrid parameter multiple body systems”. Journal of Sound and Vibration, 209(4), pp. 571–592. [12] Barhorst, A. A., 2004. “Systematic closed form modeling of hybrid parameter multiple body systems”. The International Journal of Nonlinear Mechanics, 39(1), January, pp. 63–78. [13] Barhorst, A. A., 2004. “On the efficacy of pseudocoordinates–part 1: Moving interior constraints”. The International Journal of Nonlinear Mechanics, 39(1), January, pp. 123–135. [14] Barhorst, A. A., 2004. “On the efficacy of pseudocoordinates–part 2: Moving boundary constraints”. The International Journal of Nonlinear Mechanics, 39(1), January, pp. 137–151. [15] Papastavridis, J. G., 1998. “A panoramic overview of the principles and equations of motion of advanced engineering dynamics”. Applied Mechanics Reviews, 51(4), pp. 239–265. [16] Haug, E. J., 2017. “An ordinary differential equation formulation for multiplebody dynamics: Nonholonomic constraints”. Journal of Computing and Information Science in Engineering. [17] Haug, E. J., 2016. “An index 0 differential-algebraic equation formulation for multibody dynamics: Nonholonomic constraints”. Mechanics Based Design of Structures and Machines–An International Journal. [18] Papastavridis, J., 2002. Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems : for Engineers, Physicists, and Mathematicians. Oxford University Press. [19] Kurdila, A. J., 1990. “Multibody dynamics formulations using Maggi’s approach”. In AIAA Dynamics Specialists Conference, pp. 547–558. Long Beach, CA. [20] Pars, L., 1965. Treatise on Analytical Dynamics. Heinemann.
[21] WRI, 2010-2016. Mathematica. Wolfram Research, Inc., Champaign, Illinois. [22] Barhorst, A. A., 1998. “Symbolic equation processing utilizing vector/dyad notation”. Journal of Sound and Vibration, 208(5), pp. 823–839.
