## 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* [1] to derive equations of motion that involve no constraint forces. The principle was implemented by Lagrange [2] 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 [3] and Boltzmann-Hamel [4] 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 [5] and Appell [6] 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,” [7] 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 *Gauss*’ *principle of least constraints* [8] as was shown by Appell, [9] in contrast to the Lagrange’s approach that is based on the variational *Hamilton’s principle of least action* [10] 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 [14]. 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 *holonomic system* *generalized coordinates* *holonomic generalized speeds*

where

where *generalized accelerations*. Furthermore, the velocities and angular velocities of the particles and bodies comprising a mechanical system are linear in the generalized speeds

where the generalized inertia matrix

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

Let us now consider a modification of the kinematics of

where *nonholonomic system* *nonholonomic generalized speeds* are considered to satisfy the same kinematical relations with generalized coordinates as their holonomic counterparts, i.e.,

The system dynamics of

where

where

In a similar manner, the relationships between holonomic generalized inertia forces on

Substituting (9) and (10) in (8) yields the unreduced form of Kane’s equations of motion for

The simple nonholonomic constraint equations given by (5) can be rewritten in the following matrix representation [17]:

where

where

Also, (11) can be rewritten in the matrix form [17]:

where

Notice that (17) is obtained by multiplying both sides of (4) by

## 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 *canonical set of nonholonomic generalized speeds*

where

where

where

and can be simplified further to the following form:

where

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

Time differentiating the constraint dynamics given by (19) yields [17]

where

where

where

is the projection matrix on the nullspace of

In (25), the following holds

where

## 6. Generalized accelerations from the momentum balance dynamics

Let

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

where

where

and

and

The term

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

Since

it follows that the row spaces of

Nevertheless, since [23]

then it follows from (36) that

Since the only part in the expression of

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

The same result is obtained by using the fact:

which implies by using (36) that

Since the only part in the expression of

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

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

Adopting the canonical set

In viewing the canonical generalized acceleration

Moreover, the vertical component of

and are shown to solve (23) by direct substitution and noticing that

Similarly, the horizontal component

and are shown to solve (30) by direct substitution and noticing that

Now consider a general deviation vector

The deviated canonical generalized acceleration vector

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

where

Let us now specify the deviated generalized acceleration vector

and

Substituting

which corresponds to the vertical solid red vector in Fig. (3). Notice that

where

and

where

where

Nevertheless, (59) implies that

which in terms of the square Euclidean norm implies that

Eq. (63) is exactly the statement of Gauss’ principle of least constraints [8]. The present geometric interpretation of Gauss’ principle was first introduced by Udwadia and Kalaba [24].

## 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