Open access peer-reviewed chapter

# Canonical Generalized Inversion Form of Kane’s Equations of Motion for Constrained Mechanical Systems

By Abdulrahman H. Bajodah and Ye-Hwa Chen

Submitted: November 3rd 2017Reviewed: March 20th 2018Published: July 18th 2018

DOI: 10.5772/intechopen.76648

## Abstract

The canonical generalized inversion dynamical equations of motion for ideally constrained discrete mechanical systems are introduced in the framework of Kane’s method. The canonical equations of motion employ the acceleration form of constraints and the Moore-Penrose generalized inversion-based Greville formula for general solutions of linear systems of algebraic equations. Moreover, the canonical equations of motion are explicit and nonminimal (full order) in the acceleration variables, and their derivation is made without appealing to the principle of virtual work or to Lagrange multipliers. The geometry of constrained motion is revealed by the canonical equations of motion in a clear and intuitive manner by partitioning the canonical accelerations’ column matrix into two portions: a portion that drives the mechanical system to abide by the constraints and a portion that generates the momentum balance dynamics of the mechanical system. Some geometrical perspectives of the canonical equations of motion are illustrated via vectorial geometric visualization, which leads to verifying the Gauss’ principle of least constraints and its Udwadia-Kalaba interpretation.

### Keywords

• canonical equations of motion
• discrete mechanical systems
• Kane’s method
• Gauss’ principle of least constraints
• cononical generalized speeds
• Greville formula

## 1. Introduction

Deriving mathematical models for dynamical systems is in the core of the discipline of analytical dynamics, and it is the step that precedes dynamical system’s analysis, design, and control synthesis. For discrete mechanical systems, i.e., those composed of particles and rigid bodies, the mathematical models are in the forms of differential equations or differential/algebraic equations that are derived by using fundamental laws of motion or energy principles. Because many mechanical systems nowadays are multi-bodied with numerous degrees of freedom and large numbers of holonomic and nonholonomic constraints, simplicity of the derived equations of motion is important for facilitating studying the mechanical system’s characteristics and for extracting useful information out of its mathematical model. Hence, deriving the simplest possible form of the equations of motion that govern the dynamics of the mechanical system is crucial. Moreover, because the mechanical system’s equations of motion are simulated on digital computers, computational efficiency of the derived differential equations of motion when numerically integrated is another factor by which the quality of the mathematical model is judged on.

It has been a general trend for over two centuries to employ d’Alembert’s principle of virtual work  to derive equations of motion that involve no constraint forces. The principle was implemented by Lagrange  for deriving the first set of such equations, which constituted the first paradigm shift from the Newton-Euler’s approach. The only other alternative to employing d’Alembert’s principle has been to augment the equations with undetermined multipliers, an approach that was initiated by Lagrange himself. Other formulations that followed the trend include the Maggi  and Boltzmann-Hamel  formulations. A remarkable contribution of the Lagrangian approach to analytical dynamics is utilizing the concept of generalized coordinates instead of the Cartesian coordinate concept. The choice of generalized coordinates greatly affects simplicity of the derived equations of motion.

Another paradigm shift in the subject took place when Gibbs  and Appell  independently derived their equations of motion. For the first time, formulating the dynamical equations involved neither invoking d’Alembert’s principle nor augmenting undetermined multipliers. Because d’Alembert’s principle was to many analytical dynamics practitioners, “an ill-defined, nebulous, and hence objectionable principle,”  the Gibbs-Appell model was widely accepted within the analytical dynamics community. Moreover, the absence of undetermined multipliers from the Gibbs-Appell equations contributed to maintaining simplicity and practicality of the equations for large constrained mechanical systems. Another feature of the Gibbs-Appell approach was initiating the concept of quasi-velocities, which equal in their number to the number of the degrees of freedom of the mechanical system. Similar to the advantage of generalized coordinates, carefully chosen quasi-velocities can lead to dramatic simplifications of the dynamical equations of motion.

One feature that is associated with the Gibbs-Appell’s approach is that it is based on the differential Gaussprinciple of least constraints  as was shown by Appell,  in contrast to the Lagrange’s approach that is based on the variational Hamilton’s principle of least action  as opposed. Another feature is adopting the acceleration form of constraints to model a mechanical system’s constraints. Although easy by itself, employing the acceleration form eased the historical hurdle of modeling nonholonomic constraints that used to obstruct variational-based formulations, and it is a consequence of the differential theme that is based on Gauss’ principle. In particular, the acceleration form bypassed d’Alembert’s principle and the undetermined multiplier augmentation practices that produce false equations of nonholonomically constrained motion, and it unified the treatments of holonomic and nonholonomic constraints.

A key developments in the arena of analytical dynamics is the Kane’s method for modeling constrained discrete mechanical systems [11, 12, 13]. Kane’s method adopts a vector approach that inspired useful geometric features of the derived equations of motion . The generalized active forces and generalized inertia forces are obtained by scalar (dot) multiplications of the active and inertia forces, respectively, with the vector entities partial angular velocities and partial velocities. This process delicately eliminates the contribution of constraint forces without invoking the principle of virtual work. The resulting equations are simple and effective in describing the motion of nonconservative and nonholonomic systems within the same framework, requiring neither energy methods nor Lagrange multipliers.

The standard Kane’s equations of motion for nonholonomic systems are minimal in generalized speeds, i.e., their number is equal to the number of degrees of freedom of the dynamical system, and only the independent portion of generalized speeds and their time derivatives appear in the equations. Nevertheless, information about dependent generalized speeds can be practically important, e.g., for the purpose of obtaining stability information about a dependent dynamics or when it desired to target a dependent dynamics with a control system design by using state space control methodologies.

On the other hand, generalized inversion and the Greville formula for general solutions of linear systems of algebraic equations were introduced to the subject of analytical dynamics by Udwadia and Kalaba [15, 16] as tools for deriving equations of constrained motion for discrete mechanical systems. The success that the formula met in modeling ideally constrained motion is due to its geometrical structure that captures orthogonality of ideal constraint forces on active and inertia forces, which is the essence of the principle of virtual work.

Inspired by the Udwadia-Kalaba equations of motion and the Greville formula, this chapter introduces a new form of Kane’s equations of motion. The introduced equations of motion employ the acceleration form of constraints, and therefore holonomic and nonholonomic constraints are augmented within the momentum balance formulation in a unified manner and irrespective of being linear or nonlinear in generalized coordinates and generalized speeds. The equations of motion are nonminimal, i.e., no reduction of generalized speed’s space dimensionality takes place from the number of generalized coordinates to the number of degrees of freedom. Furthermore, the new equations of motion are explicit, i.e., are separated in the generalized acceleration variables, and only one generalized acceleration variable appears in each equation.

The main feature of the derived equations of motion is the explicit algebraic and geometric partitioning of the generalized acceleration vector at every instant of time into two portions: one portion drives the mechanical system to abide by the constraint dynamics, and the other portion generates the momentum balance of the mechanical system as to follow Newton-Euler’s laws of motion.

## 2. Kane’s equations of motion for holonomic systems

Consider a set of νparticles and μrigid bodies that form a holonomic system Shpossessing ndegrees of freedom in an inertial reference frame J. Assume that ngeneralized coordinates q1,,qnare used to describe the configuration of the system. Then a corresponding set of nholonomic generalized speeds uh1,,uhnis used to model the kinematics of the system. The two sets are related by the kinematical differential equations [12, 13]:

q̇=Cqtuh+Dqt,E1

where qRnis a column matrix containing the generalized coordinates; uhRnis a column matrix containing the generalized speeds, q̇=dq/dt, CRn×n, DRn; and C1exists for all qRnand all tR[12, 13]. Kane’s dynamical equations of motion for Share given by [12, 13]

Frquht+Frquhu̇ht=0,r=1,,n,E2

where Frand Frare the rthholonomic generalized active force and the rthholonomic generalized inertia force on the system, respectively, and u̇h=duh/dtRnis a column matrix containing the generalized accelerations. Furthermore, the velocities and angular velocities of the particles and bodies comprising a mechanical system are linear in the generalized speeds uhr. Hence, the accelerations, angular accelerations, and consequently the generalized inertia forces are linear in the generalized accelerations u̇hr. Therefore, a column matrix FRncontaining Fr,r=1,,ncan be written in the following form :

Fquhu̇ht=Qqtu̇hLquht,E3

where the generalized inertia matrix QRn×nis assumed symmetric and positive definite and LRn. Hence, a matrix form of (2) is written as 

Qqtu̇h=Lquht+Fquht.E4

## 3. Kane’s equations of motion for nonholonomic systems

Let us now consider a modification of the kinematics of Shthat is made by imposing the following simple nonholonomic constraints on the generalized speeds [12, 13]:

up+r=s=1pArsqtus+Brqt,r=1,,m,E5

where u1,,unare the generalized speeds of the nonholonomic system Sthat is resulting from constraining Shaccording to (5), m=np, and Arsand Brare scalar functions of the generalized coordinates q1,,qn,and t. The nonholonomic generalized speeds are considered to satisfy the same kinematical relations with generalized coordinates as their holonomic counterparts, i.e.,

q̇=Cqtu+Dqt.E6

The system dynamics of Schanges from that given by (2) accordingly. Let the generalized speed column matrix be partitioned as

u=uITuDTT,E7

where uI=u1upTand uD=up+1unT. Kane’s dynamical equations of motion for Sare given by [12, 13]

FrquIt+FrquIu̇It=0,r=1,,p,E8

where Frand Frare the rthnonholonomic generalized active force and the rthnonholonomic generalized inertia force on S, respectively. The relationships between holonomic generalized active forces on Shand nonholonomic generalized active forces on Sare given by [12, 13]

FrquIt=Frqut+s=1mFp+squtAsr,r=1,,p.E9

In a similar manner, the relationships between holonomic generalized inertia forces on Shand nonholonomic generalized inertia forces on Sare given by [12, 13]

FrquIu̇It=Frquu̇t+s=1mFp+squu̇tAsrqt,r=1,,p.E10

Substituting (9) and (10) in (8) yields the unreduced form of Kane’s equations of motion for S[12, 13, 17]:

Frqut+Frquu̇t+s=1mFp+squt+Fp+s(quu̇t)Asrqt=0,r=1,,p.E11

The simple nonholonomic constraint equations given by (5) can be rewritten in the following matrix representation :

uD=AqtuI+Bqt,E12

where ARm×pand BRm. Furthermore, (12) can be rewritten as 

A1qtu=Bqt,E13

where A1Rm×nis given by

A1qt=AqtIm×m.E14

Also, (11) can be rewritten in the matrix form :

A2qtFquu̇t=A2qtFqut,E15

where A2Rp×nis given by

A2qt=Ip×pATqt.E16

Hence, (15) becomes 

A2qtQqtu̇=A2qtLqut+A2qtFqut.E17

Notice that (17) is obtained by multiplying both sides of (4) by A2qt. Therefore, the unique holonomic generalized acceleration vector u̇hthat solves the fully determined system given by (4) solves the underdetermined system given by (17) also, among an infinite number of generalized acceleration vectors that satisfy (17), each of which preserves a constrained momentum balance dynamics of the mechanical system.

## 4. Canonical generalized speeds

Choosing the set of generalized speeds is a crucial step in formulating Kane’s dynamical Eqs. (2) and (8) because the extent of how complex these equations appear is affected by this choice. For every choice of nonholonomic generalized speeds u1,,un, we define the canonical set of nonholonomic generalized speeds w1,,wnsuch that

w=Q1/2qtu,E18

where wis the column matrix containing w1,,wnand Q1/2is the square root matrix of Q. With this choice of generalized speeds, (13) becomes

A1qtw=Bqt,E19

where A1qt=A1qtQ1/2qt. The time derivative of (18) is

ẇ=Q̇1/2qutu+Q1/2qtu̇E20

where Q̇1/2is the element-wise time derivative of Q1/2along the trajectory solutions of the kinematical differential Eqs. (6). Therefore, (17) becomes

A2qtQ1/2qtẇ=A2qtLqut+A2qtFqut+A2qtQ1/2qtQ̇1/2qutuE21

and can be simplified further to the following form:

A2qtẇ=A2qtQ1/2qtFqutLqut+A2qtQ̇1/2qutuE22

where A2qt=A2qtQ1/2qt. We view the nonholonomic mechanical system dynamics as being composed of two parts: a constraint dynamics that is modeled by (19) and a momentum balance dynamics that is modeled by (22). Scaling velocity variables and constraint matrices by square roots of the inertia matrices for the purpose of characterizing constrained motion is implicit in Gauss’ principle of least constraints  as will be shown later in this paper. Moreover, deriving explicit equations of motion for constrained mechanical systems by utilizing this type of scaling is first due to Udwadia and Kalaba. [15, 16] The arguments of the functions are omitted in the remaining sections for brevity, unless necessary to clarify concepts.

## 5. Generalized accelerations from the acceleration form of constraints

Time differentiating the constraint dynamics given by (19) yields 

A1ẇ=V1,E23

where V1Rmis given by

V1=ḂqutȦ1qutw,E24

where Ḃand Ȧ1are the element-wise time derivatives of Band A1along the trajectory solutions of the kinematical differential Eqs. (6). The general solution of the above-written acceleration form of constraint equations for ẇis given by the Greville formula as [18, 19, 20]

ẇ=A1+V1+P1y1,E25

where A1+is the Moore-Penrose generalized inverse (MPGI) [21, 22] of A1and

P1=In×nA1+A1E26

is the projection matrix on the nullspace of A1and y1Rnis an arbitrary vector as for satisfying the acceleration form given by (25) but is yet to be determined to obtain the unique natural generalized acceleration. Because Q1/2is of full rank, it follows that A1retains the full row rank of A1and hence that A1+Rn×mis given by the closed form expression:

A1+=A1TA1A1T1.E27

In (25), the following holds

A1+V1RA1T,P1y1NA1E28

where Rand Nrefer to the range space and the nullspace, respectively. The term A1+V1in (25) is the minimum norm solution of (23) for ẇamong infinitely many solutions that are parameterized by y1.

## 6. Generalized accelerations from the momentum balance dynamics

Let V2Rnbe given by

V2=Q1/2FL+Q̇1/2u.E29

Then the momentum balance Eq. (22) takes the following compact form:

A2ẇ=A2V2E30

where A2retains the full row rank of A2because Q1/2is of full row rank. Hence, another expression for the general solution of ẇis obtained by utilizing the Greville formula to solve (30) and is given by

ẇ=A2+A2V2+P2y2E31

where A2+Rn×pis given by the closed form expression:

A2+=A2TA2A2T1E32

and

P2=In×nA2+A2E33

and y2Rnis an arbitrary vector as for satisfying the momentum balance dynamics given by (30), but its unique value that solves for the natural generalized acceleration vector ẇis yet to be determined, and

A2+A2V2RA2+A2=RA2T,P2y2NA2.E34

The term A2+A2V2in (31) is the minimum norm solution of (30) for ẇamong infinitely many solutions that are parameterized by y2.

## 7. Canonical generalized inversion Kane’s equations of motion

Since A1and A2are full row rank matrices and their numbers of rows mand psum up to the full space dimension nand since

A1A2T=A1Q1/2A2Q1/2T=A1Q1/2A2Q1/2T=A1Q1/2Q1/2A2T=A1A2T=A+A=0m×mE35

it follows that the row spaces of A1and A2are orthogonal complements, i.e.,

RA1T=RA2T.E36

Nevertheless, since 

RA1T=NA1E37

then it follows from (36) that

RA2T=NA1.E38

Since the only part in the expression of ẇgiven by (25) that is in NA1is the second term P1y1, and since the ony part in the equivalent expression of ẇgiven by (31) that is in RA2Tis the first term A2+A2V2, it follows from (38) that

P1y1=A2+A2V2.E39

Substituting (39) in (25) yields the canonical generalized inversion form of Kane’s equations for nonholonomic systems:

ẇ=A1+V1+A2+A2V2.E40

The same result is obtained by using the fact:

NA2=RA2TE41

which implies by using (36) that

RA1T=NA2E42

Since the only part in the expression of ẇgiven by (25) that is in RA1Tis the first term A1+V1, and since the only part in the equivalent expression of ẇgiven by (31) that is in NA2is the second term P2y2, it follows from (42) that

P2y2=A1+V1.E43

(Substituting (43) in (31) yields Eq. (40). Eq. (20) can be used to express (40) in terms of the original generalized acceleration vector u̇, resulting in

u̇=Q1/2A1+V1+A2+A2V2Q̇1/2u.E44

## 8. Geometric interpretation of the canonical generalized inversion form

Adopting the canonical set w1,,wnof generalized speeds in deriving the dynamical equations for a mechanical system reveals the geometry of its constrained motion. Figure 1 depicts a geometrical visualization of the ndimensional Euclidian space at an arbitrary time instant t. The vertical and the horizontal axes resemble the orthogonally complements mdimensional and pdimensional subspaces RA1Tand RA2T, respectively.

In viewing the canonical generalized acceleration ẇgiven by (40) as the geometrical vector shown in Figure 1, it is shown to be composed of two components that are orthogonal to each other: The vertical component A1+V1resides in RA1T, and it enforces the constraint dynamics given by (23), and the horizontal component A2+A2V2resides in RA2T, and it generates the momentum balance dynamics given by (30).

Moreover, the vertical component of ẇis the shortest in length “minimum norm” solution among infinitely many solutions of (23) that are parameterized by y1according to (25). These solutions can also be represented by arbitrary horizontal deviation vectors: Δ2iẇ=A2+A2δiV2RA2T,i=1,2,as

ẇ+Δ2iẇ=A1+V1+A2+A2V2+δiV2E45

and are shown to solve (23) by direct substitution and noticing that A1A1+=Im×mand A1A2+=0m×p. Two of these solutions are plotted (in dotted red) in Figure 1 for arbitrary vectors δ1V2and δ2V2in Rn, in addition to the natural generalized acceleration vector ẇthat is obtained by setting δiV2=0n.

Similarly, the horizontal component A2A2+V2of ẇis the shortest solution among infinitely many solutions of (30) that are parameterized by y2according to (31). These solutions can also be represented by arbitrary vertical deviation vectors: Δ1iẇ=A1+δiV1RA1T,i=1,2,as

ẇ+Δ1iẇ=A2+A2V2+A1+V1+δiV1E46

and are shown to solve (30) by direct substitution and noticing that A2A2+=Ip×pand A2A1+=0p×m. Two of these solutions are plotted (in dotted blue) in Figure 1 for arbitrary vectors δ1V1and δ2V1in Rm, in addition to the natural generalized acceleration vector ẇthat is obtained by setting δiV1=0m. Notice that the canonical generalized acceleration vector ẇis the only solution that solves (45, 46) simultaneously and is obtained by setting δiV2=0nand δiV1=0m.

Now consider a general deviation vector Δẇthat is composed of arbitrary vertical and horizontal deviation components from ẇas shown in Figure 2. The vertical component A1+δV1abides by (46) but violates (45), and the horizontal component A2+A2δV2abides by (45) but violates (46). Hence:

Δẇ=A1+δV1+A2+A2δV2.E47

The deviated canonical generalized acceleration vector ẇ+Δẇis obtained by summing (40) and (47) as

ẇ+Δẇ=A1+V1+δV1+A2+A2V2+δV2E48

and is shown in Figure 2 in dotted blue. On the other hand, the canonical holonomic generalized acceleration vector in terms of the canonical generalized speeds is obtained from (4) and (20) as

ẇh=V2E49

where wh=Q1/2uand usolves (4). Decomposing the expression of ẇhalong RA1and RA2yields

ẇh=V2=P1V2+P2V2E50
=In×nA1+A1V2+In×nA2+A2V2E51
=A2+A2V2+A1+A1V2.E52

Let us now specify the deviated generalized acceleration vector ẇ+Δẇto be ẇhas shown in Figure 3. Equating the two expressions (48) and (52) and solving for δV1and δV2yield

δV1=A1V2V1,E53

and

δV2=0n.E54

Substituting δV1and δV2in (47) yields

Δẇ=A1+A1V2V1E55

which corresponds to the vertical solid red vector in Fig. (3). Notice that Δẇis the shortest among all deviation vectors that end up at ẇh(two of which are shown in dotted red) by deviating from generalized acceleration vectors that abide by the constraint dynamics given by (23) (two of which are shown in dotted green), i.e.,

Δẇ=ẇhẇ=miniΔẇi=miniẇhẇi,i=1,2,E56

where ẇisatisfies

A1ẇi=V1E57

and xdenotes the Euclidean norm of xgiven by x=xTx. Moreover, Δwcan be expressed in terms of the original set of generalized speeds as

Δw=Q1/2ΔuE58

where Δu=uhuis the difference between holonomic and nonholonomic generalized speeds. Therefore:

Δẇ=Q1/2Δu̇+Q̇1/2Δu=miniQ1/2Δu̇i+Q̇1/2Δu,i=1,2,E59

where Δu̇i=u̇hu̇iand u̇isatisfies

A1ẇi=A1Q1/2u̇i+Q̇1/2u=V1.E60

Nevertheless, (59) implies that

Q1/2Δu̇=miniQ1/2Δu̇i,i=1,2,.,E61

which in terms of the square Euclidean norm implies that

Q1/2Δu̇2=Δu̇TQ1/2Q1/2Δu̇=miniΔu̇iTQ1/2Q1/2Δu̇iE62
=Δu̇TQΔu̇=miniΔu̇iTQΔu̇i,i=1,2,.E63

Eq. (63) is exactly the statement of Gauss’ principle of least constraints . The present geometric interpretation of Gauss’ principle was first introduced by Udwadia and Kalaba .

## 9. Conclusion

The chapter introduces the canonical generalized inversion dynamical equations of motion for nonholonomic mechanical systems in the framework of Kane’s method. The introduced equations of motion use the Greville formula and utilize its geometric structure to produce a full order set of dynamical equations for the nonholonomic system. Moreover, the acceleration form of constraint equations is adopted in a similar manner as in the classical Gibbs-Appell, Udwadia-Kalaba, and Bajodah-Hodges-Chen formulations.

The philosophy on which the present formulation of the dynamical equations of motion is based views the constrained system dynamics of the mechanical system as being composed of a constraint dynamics and a momentum balance dynamics that is unaltered by augmenting the constraints. Inverting both dynamics by means of two Greville formulae and invoking the geometric relations between the resulting two expressions yield the unique natural canonical generalized acceleration vector.

Because the momentum balance dynamics and the acceleration form of constraint dynamics are linear in generalized accelerations, only linear geometric and algebraic mathematical tools are needed to analyze constrained motion of discrete mechanical systems. Also, the present linear analysis is valid in despite of dependencies among the constraint equations and changes in rank that the constraint matrix Amay experience because the matrices A1and A2are always of full row ranks and their mand prows span two orthogonally complement row spaces. Another advantage of maintaining full row ranks of A1and A2is that their generalized inverses have explicit and closed form expressions, which alleviate the need for employing numerical methods for computing generalized inverses.

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Abdulrahman H. Bajodah and Ye-Hwa Chen (July 18th 2018). Canonical Generalized Inversion Form of Kane’s Equations of Motion for Constrained Mechanical Systems, Nonlinear Systems - Modeling, Estimation, and Stability, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/intechopen.76648. Available from:

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