Instantaneous Kinematics (IK)

Instantaneous Kinematics, IK, is helpful when you need a mechanism for a particular design application.

For example, you can use IK to help you search for:

Cusps: sometimes called a 'pick-up motion'.

A coupler-curve that comes to a Point. The coupler point is on the moving polode, such that it approaches the fixed polode and 'kisses' it (osculates), and then reverses away from it. It is generated when the coupler point coincides with the instantaneous centre of rotation of the coupler.

Crunodes: Double Points

A coupler-curve that passes through the same point two times, but with a different tangent. The coupler-curve has a loop, or it a figure-of-eight.


A coupler-curve that is a straight line for a substantial section. The line is approximate or exact.

Constant Radius:

A coupler-curve that is a constant radius for a substantial section

Symmetrical Coupler Curves:

A coupler-curve that is symmetric about an axis line. These can be used for indexing mechanisms and, by adding two more links, to give a mechanism with two dwells.



(In this topic, I am using the Mechanism Synthesis software, from University of Basque Country, Spain. I am using their software to show the 'classic' instantaneous curves that are often very useful for mechanism synthesis.

I must acknowledge Alfonso Hernández, CompMech, Department of Mechanical Engineering, UPVEHU for the permission to use the GIM® software. (


Grashof - See Grashof Criterion

The length of the shortest part plus the longest Part of a planar four-bar kinematic-chains cannot be greater than the lengths of the other two parts if there is to be continuous relative motion between the Parts.

Or, when a minimum of one part rotates, the kinematic-chain is called a Grashof.

With the definitions for:

S (length of the shortest Part), L (length of the longest Part), P and Q the length of the other two Parts
If L+S > P+Q then the kinematic-chain is a Non-Grashof.
If L+S < P+Q then the kinematic-chain is a Grashof.


The Instantaneous Centres (IC) - Pole

The orange dot , I14, is the Instantaneous Centre (IC) of rotation of the coupler relative to the fixed frame.

We also call this IC the 'Pole', or the 'Velocity Pole'.

It is at the intersection of two rays. One ray is the along the axis of the 'crank', and the other ray is along the axis of the 'rocker'. Clearly, the Pole,  I14, moves as the mechanism cycles.

There are five other Instantaneous Centres.

Those ICs at each of the four joints.
One more that is at the intersection of the ray along the axis of the coupler with the ray along the axis of the frame.


Fixed Polode

Since the IC Pole ,,moves, we can map the Pole on to the Frame at each step of the Crank rotation.

The trace of the Pole on the Frame is the Fixed Polode.

We cannot see the Fixed Polode - we must either draw it or imagine it.

The purple trace is the Fixed Polode.

Notice the Fixed Polode passes through the centre of the rocker two times



Moving Polode

We can also map the IC Pole on to the Coupler at each step of the Crank rotation.

The trace of the Pole on to the Coupler is the Moving Polode.

We cannot see the Moving Polode - we must either draw it or imagine it.

The green trace is the Moving Polode.

Notice the Moving Polode passes through the end of the rocker two times.



Fixed and Moving Polode

When we show the Fixed and Moving Polode you can see that they are always in continuous contact at the Pole I14 as the mechanism makes a cycle.

Why? At each instant, the Pole:

is the Point mapped on to the Frame - a point on the Fixed Polode  ...

...and also...

... is the Point mapped on to the Coupler - a point on the Moving Polode

Since it is the same Point, the Fixed and Moving Polodes are in contact - always.


Pole Tangent and Pole Normal.

Because the Moving Polode is always is in contact with the Fixed Polode at the position of the Pole, they must have a common tangent line at the Pole.

Why a Tangent?

Logically, if the Polodes were at an angle, that is, they were not tangent with each other at their contact Point, then, at the next instant, they would not be in contact.

The Tangent to the Moving and Fixed Polode is called the Pole Tangent.
The Line that is Normal () to the Pole Tangent is called the Pole Normal

The Moving Polode and the Fixed Polode 'Roll' along each other without slipping.


Coupler Curves

A coupler-curve is the trace that a point on the coupler (the Coupler Point, K) makes on the fixed frame.

Each point on the coupler will trace out a different coupler-curve.


Are there interesting coupler-curves? FOR THIS PARTICULAR FOUR-BAR.


Make the Point on the Coupler coincide with any point along the the Moving Polode. At some other phase in the mechanism cycle, the Coupler-Curve with become a Cusp when the Point that is on the Moving-Polode becomes in contact with the Fixed-Polode.

In this image, the Coupler Point, K, is on the Moving Polode.

As the four-bar cycles, the Couple-Point K, approaches the Fixed-Polode and touches it at a Cusp Point. The motion of the Point is perpendicular () to the Fixed-Polode.



Moving a Coupler Point along the Moving Polode to show that the Coupler Point moves towards the Fixed Polode with a Cusp

You can see in the Video as I move the Position of the Coupler Point along the Moving Polode, then Coupler Point will move along the Fixed Plane until it touches the Fixed Polode.


Question: Is there anything interesting about the Points along the Coupler Curve.

Answer: Yes!

The motion of the Coupler Point, K, is always perpendicular () to the IC Pole I14

Why is this the case?

Because the Pole I14 is the centre of rotation of the Coupler Plane.

The instantaneous motion of every other point (K in the image to the left) on the coupler moves moves in a perpendicular direction to the ray drawn from the Pole I14.

Notice, also, that the Pole has a Velocity in the same direction of the Pole Tangent.




Osculating Circle

The Coupler Curve always has a Curvature, and, thus, a 'Radius-of-Curvature'.

We can draw a 'circle' at the Coupler Point, K.

When the circle has the same radius as the Radius-of-Curvature of the Coupler Curve, at the Coupler-Point, K, we call the Circle the Osculating Circle

It is of interest that the:

Coupler Point, K
Centre of the Osculating Circle, R
Pole, I14

...are always along a single ray.

A ray joins the Pole, I14, the Centre of the Osculating Circle, R,  and the Coupler Point, K.

More Detailed Definition of Osculating Curve

There is intimate contact between Osculating Circle and the Coupler Curve with three infinitesimally separated points. (imagine a point to each sides of K, called -K, and +K, on the Coupler Curve. Draw Perpendicular Bisectors between the secants -KK and K+K. These will intersect at the centre of the Osculating Circle in the limit as -K, K and +K become close together. Point -K, K and +K are the three infinitesimally separated Points.


Coupler Curves in which the Radius of Curvature at the Coupler-Point is Infinite.

Question: Why might this be interesting?

Answer: Because a Coupler Curve that has an infinite radius is in fact a straight line.

There is a Circle of Points that move in a straight line for each position of the Mechanism.

The circle is called the Inflection Circle.

All of the Points in the Cyan Circle in this image move along a straight line

Remember, this is the Inflection Circle only for this four-bar at this instant in the cycle.

Everything is always changing.

How do we calculate the diameter and the centre of the inflection circle?



The Euler-Savary Equation

Assume you have a four-bar mechanism, and you want to find the inflection circle.

We can find the Pole. The Pole lies on the Inflection Circle.

The Euler-Savary Equation...

1/I.A - 1/I.A0 = 1/I.JA

We know distance I.A - it is the distance from the Pole to the End of the Crank

We know distance I.A0 - it is the distance from the Pole to the Centre of the Crank

We can calculate the distance I.JA - the distance from the Pole to the Point on the inflection circle.

All three distances lie on the same ray.


There is a different way to find the Pole Tangent

Draw a Line from the Instant Centre I14 to I13

This is called the Collineation Axis

Note the Angle β between the Line from I14 to the Centre of Rotation of the Crank

The angle β is the same to the Pole Tangent from the Line I14 to the Centre of Rotation of the Rocker

You can then draw in the Pole Normal - Not shown since, by chance it is close to the Collineation Axis.




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