## The Planar Euler-Savary Equation in Vectorial Notation

### Stefan Gössner

*Department of Mechanical Engineering, University of Applied Sciences, Dortmund, Germany.*

**Keywords**: Euler-Savary Equation; pole transfer velocity; inflection Pole; inflection circle; rho-curves; cubic curve of stationary curvature; Ball's point; undulation point; geometric kinematics;

## Abstract

The Euler-Savary Equation is discussed from a vectorial point of view. Kinematic properties of a moving link in the plane are taken to derive the inflection circle first. Then eliminating all kinematic values results in the pure geometric equation of Euler-Savary. Proceeding to the advantageous canonical coordinate system the *curves of points of constant curvature*, the *cubic of stationary curvature* and *Ball's point* location are derived.

## Introduction

In mechanism analysis and design knowledge about direction and curvature of point paths on a moving link of kinematic chains proves itself valuable and was intensively studied in the past [1,2,3,4]. In this article the kinematic properties of points on a moving plane are discussed first. From here pure geometric relations will be derived:

- Inflection Pole and Circle
- The Curve of Points of Constant Curvature
- The Cubic of Stationary Curvature
- Ball's Point

The famous Euler-Savary equation is a central point in the discussion of these properties. This equation is derived here based on a vectorial notation using an orthogonal operator [5,6]. This leads to a general-purpose result, which is independent of the commonly used canonical coordinate system.

## Kinematics

Consider a moving plane with known *velocity pole*

These two points are called *conjugate points*, which are lying together with the velocity pole *pole ray*. The velocity

as the pole has no velocity by definition. The acceleration

with pole acceleration

and deriving it again w.r.t. time yields

Herein is *pole transfer velocity*

The direction of the

pole transfer velocity$\\bold u$ coincides with the direction of thepole tangent$t$ , whereas thepole acceleration$\\bold a\_P$ coincides with the direction of thepole normal$n$ .

We can interprete velocity and normal acceleration of point

From these two expressions

That term is introduced back into (5), while being allowed to write

Equation (1) in its form

## Inflection Circle

We now want to have a closer look at points on the moving plane, which are inflection points of their path at current. Those are points at which their curve changes from being *concave* to *convex*, or vice versa, so their radius of curvature is instantaniously infinite. Such a point - say

Reusing equations (1) and (2) gives

and resolving the brackets leads to the quadratics

Completing the square results in

which has the shape

All points on a moving plane, that are inflection points of their path at current, are located on a circle - the

inflection circle.

The pole *inflection pole*

## The Euler-Savary Equation

Substituting

Multiplication of equation (6) with

which can be resolved for

Now reuse of the terms *Euler-Savary equation*

The Euler-Savary equation (9) associates conjugate points of a moving plane with the relative location of velocity pole and inflection pole in a pure geometrical form.

The intersection point

so Euler-Savary equation can be also written as

If we happen to know two pairs of conjugate points

According to (10) they are

and with these we are able to write down the Euler-Savary equation two times

Now we can synthesize a geometric vectorial equation for the location of the inflection pole

With the knowledge of two pairs of conjugate points of a moving plane, the location of the reflection pole

$W$ can be determined by equation (12).

Equation (12) is a vectorial alternative to *Bobillier's construction* of the inflection pole.

If we know the inflection pole's location, we can also calculate to any point

Equation (13) is the geometric pendant to kinematic equation (6). When the denominator in (13) becomes zero, the radius of curvature of the path of

is a pure geometric, necessary condition for any point

## The Curvature of Point Paths

In order to investigate the curvature of the path of a point

Then we call

In this *canonical system* Euler-Savary equation (9) reads

Using

Introducing

which - after inverting - results in the well known scalar Euler-Savary equation

Herein *minus* in the parentheses has to be changed to a *plus*. Using the vectorial form (9), we have the comfort to work with an arbitrary reference coordinate system and don't need to care about signs. Resolving equation (14) for the curvature radius

which Freudenstein calls the *quadratic form* of the Euler-Savary equation [8].

The transformation from an arbitrary user coordinate system to the canonical one aligned with pole tangent and pole normal requires the orientation of

vectors are transformed via

## The Curve of Points of Constant Curvature

All points of a moving plane, possessing the same radius of curvature * $\\rho$-curve*. We get it by resolving equation (16) for

The curve of corresponding centers of curvature results from substituting

The

The plus sign before the square root was used in equations (18) and (19). The alternative curves (negative sign) results in mirrored curves with respect to the pole tangent.

When the given curvature radius goes to infinity, the

## The Cubic of Stationary Curvature

Some points on the moving plane possess a stationary curvature, i.e. their rate of change of their curvature radius

As a result we get the *cubic curve of centers of stationary curvature*. See [1-4, 7-10] for a more indepth discussion.

All we need to know about coefficents

and then solve for

Reintroducing those coefficients into (20), while inverting, yields the cubic equation of all points of the moving plane in polar vector notation

The plot of that curve in Fig. 6 shows all points of the fourbar coupler with stationary curvature of their paths. The cubic of stationary curvature belongs to a family of curves mathematically termed *strophoids*.
When the denominator in equation (21) approaches zero, i.e.

With certain poses of the fourbar mechanism one of the constants

For plotting the curve in practice, one has to calculate the coordinates

## Ball's Point

All points on the cubic possess stationary curvature and all points on the inflection circle are running through an inflection point of their paths, so having instantaneous infinite curvature. Thus the intersection point of the cubic curve with the inflection circle has an important property of stationary infinite curvature of fourth order. That point is called *Ball's point* and is in fact a *point of undulation*, where the curvature vanishes, but does not change sign.

The equation of the inflection circle in canonical polar coordinates reads

Equating it with (21) gives us the angular location of Ball's point

which has been marked also in Fig. 6.

## Conclusion

A discussion of the path curvature of points on a moving plane using a vectorial approach could be done with comparable low effort. The resulting equations prove valuable for practical engineering applications and for visualization via computer graphics.

## References

[1] O. Bottema, B. Roth, Theoretical Kinematics, Dover, 1979

[2] R.S. Hartenberg, J. Denavit, Kinematic Synthesis of Linkages, McGraw-Hill, 1964

[3] J.J. Uicker et al., Theory of Machines and Mechanisms. Oxford Press, 2011

[4] M. Husty et al., Kinematik und Robotik. Springer, 1997

[5] S. Gössner, Mechanismentechnik – Vektorielle Analyse ebener Mechanismen, Logos, Berlin, 2016

[6] S. Gössner, Cross Product Considered Harmful.

[7] B. Dizioglu, Getriebelehre - Grundlagen, Vieweg Verlag, 1965.

[8] F. Freundenstein, G. Sandor, Mechanical Design Handbook - Kinematic of Mechanisms, McGraw-Hill, 2006.

[9] W. Blaschke, H.R. Müller, Ebene Kinematik, Oldenbourg Verlasg, 1965.

[10] H. Stachel, Strophoids – Cubic Curves with Remarkable Properties, 24th Symposium on Computer Geometry SCG 2015.