Instant centre of rotation

The instant centre of rotation, also called "instantaneous centre", for a plane figure moving in a two dimensional plane is a point in its plane around which all other points on the figure, for one instant, are rotating. This point itself is the only point that is not moving at that instant.

Analysis

When a figure is moving in a plane from position 1 to position 2 it is subject to a combination of rotation and translation. However, a point may be determined around which the figure has virtually "rotated". That point, called instant centre of rotation, is not moving and exists for just one instant, for when the figure continues to move a new instant centre of rotation may be determined.

To determine the instant centre of rotation one needs to choose only two points on the surface of the figure, in this case point A and point B. See sketch 1. The figure is defined in position 1 by A¹ and B¹ and in position 2 by A² and B². It may be observed from the sketch that the figure must have rotated ánd translated to move from position 1 to position 2.

The next step requires bisecting the line A¹-A². Any point on this bisection may be the centre of a circle where on its circumference the points A¹ and A² are located. Also any point on the bisection of the line B¹-B² may be the centre of a circle where on its circumference the points B¹ and B² are located. Where these two bisections cross that is the one point that itself is not moving and that is the centre of two concentric circles on which these points are located. This one point is the instant centre of rotation P for these two positions.

Extremes

For a plane figure that is solely subject to translation –rotation is zero– the instant centre of rotation is located in infinity. This instant centre of rotation moves along in the same direction as the one the figure is moving.
When there is no translation, and all points of the plane figure only rotate around a point that does not move than that point is the centre of rotation.

Centrodes

Since the figure continues to move, any following instant a new instant centre of rotation may be determined. The result is a series of instant centres of rotation that together result in a curve. The appearance of the curve depends on the point of observation. Relative to the inertial frame of reference this curve is called a space-centrode. Relative to the rotating body the curve is called a body-centrode.

Explanation using a simple wheel, observed from a position on the road

A wheel rolls on a road, see sketch 2. Although the wheel is rotating around its axis M, the axis with the wheel also moves forward. P is the point where the wheel touches the road surface. Assuming no slippage the speed of P is zero. Since the wheel rolls forward all points on the wheel move except point P, at that particular instant. Therefore, P is at that moment the instant centre of rotation for that wheel.

Every next instant centre of rotation P is also located on the wheel-road interface. Therefore, the space-centrode follows the road surface.

All points on the wheel move with a constant angular velocity around the instant centre of rotation P. All these points on the wheel are located on the circumference of concentric circles with P at the centre of these circles. Therefore, every point on the wheel is connected with a radius to point P. The "direction of motion" of a point on the wheel is perpendicular to that radius (which coïncides with the tangent to the circle). The direction of motion and size of speed of a number of points are illustrated with a vector for the points in sketch 2.

The further away from point P the proportionally larger the speed. Therefore, the point at the top of the wheel moves in the same direction as the centre M of the wheel, but twice as fast, since it is twice the distance away from P. All points on distance 'r' from point P move at the same speed M does but in different directions. This is shown for one point in the right bottom corner: it has the same speed as M but it moves in a different direction than M does, tangential to the circle.

Instant centre of rotation and mechanisms

Sketch 3 shows a four bar linkage where a number of instant centres of rotation are illustrated. The rigid body noted by the letters BAC is connected with links P₁-A and P₂-B to a base or frame.
The three moving parts of this mechanism (the base is not moving) are: link P₁-A, link P₂-B, and body BAC. For each of these three parts an instant centre of rotation may be determined.

Considering first link P₁-A: all points on this link, including point A, rotate around point P₁. Since P₁ is the only point not moving in the given plane it may be called the instant centre of rotation for this link. Point A, at distance P₁-A from P₁, moves in a circular motion in a direction perpendicular to the link P₁-A, as indicated by vector V_A.
The same applies to link P₂-B: point P₂ is the instant centre of rotation for this link and point B moves in the direction as indicated by vector V_B.

For determining the instant centre of rotation of the third element of the linkage, the body BAC, the two points A and B are used because its moving characteristics are known, as derived from the information about the links P₁-A and P₂-B.
The direction of speed of point A is indicated by vector V_A. Its instant centre of rotation must be perpendicular to this vector (as V_A is tangentially located on the circumference of a circle). The only line that fills the requirement is a line colinear with link P₁-A. Somewhere on this line there is a point P, the instant centre of rotation for the body BAC.
What applies to point A also applies to point B, therefore this instant centre of rotation P is located on a line perpendicular to vector V_B, a line colinear with link P₂-B. Therefore, the instant centre of rotation P of body BAC is the point where the lines through P₁-A and P₂-B cross.

Since this instant centre of rotation P is the centre for all points on the body BAC for any random point, say point C, the speed and direction of movement may be determined: connect P to C. The direction of movement of point C is perpendicular to this connection. The speed is proportional to the distance to point P.

Continuing this approach with the two links P₁-A and P₂-B rotating around their own instant centres of rotation the centrode for instant centre of rotation P may be determined. From this the path of movement for C or any other point on body BAC may be determined.

Examples of application

In biomechanical research the instant centre of rotation is observed for the functioning of the joints in the upper and lower extremities ( [http://muscle.ucsd.edu/musintro/ma.shtml] ).For example in analysing the knee joint: [http://civil-ws2.wpi.edu/Documents/Roadsafe/KTH/Biblio/Docs/Knee%20joint%20motion%20description%20and%20measurement.pdf] , [http://www.ncbi.nlm.nih.gov/pubmed/12897612?ordinalpos=8&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVDocSum] , [http://www.ncbi.nlm.nih.gov/pubmed/12893038?ordinalpos=12&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVDocSum] , the ankle joint: [http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2231068] , or the shoulder joint: [http://www.uwinnipeg.ca/faculty/pass/kah/faculty/jpeeler/Ergonomics/Microsoft%20PowerPoint%20-%20Biomechanics%20of%20shoulder.ppt.pdf] and [http://www.ncbi.nlm.nih.gov/pubmed/1254624?ordinalpos=38&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVDocSum] .Such knowledge assists in developing artificial joints and prosthesis, such as the elbow: [http://www.patentstorm.us/patents/5030237/description.html] or the finger joints: [https://ozone.scholarsportal.info/bitstream/1873/3691/1/245277.pdf] .

Study of the joints of horses: "...velocity vectors determined from the instant centers of rotation indicated that the joint surfaces slide on each other." [http://www.ncbi.nlm.nih.gov/pubmed/3223666?ordinalpos=16&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVDocSum] .

Studies on turning a vessel moving through water: [http://www.puertos.es/export/download/ROM_PDFs/Rom31_99_PART_VI.pdf] .

The braking characteristiscs of a car may be improved by varying the design of a brake pedal mechanism: [http://www.wikipatents.com/gb/1443270.html] .

Designing the suspension of a bicycle: [http://www.patentstorm.us/patents/7100930/claims.html] , or of a car: [http://books.google.com/books?id=Pvsv78xj7UIC&pg=PA346&lpg=PA346&dq=%22Instant+center+of+rotation%22+suspension+coupler&source=web&ots=1mzxlJhHhq&sig=O5dtrd6vzgJxGMERXNH1XolOIrw&hl=en&sa=X&oi=book_result&resnum=1&ct=result#PPA348,M1] .

In the case of the coupler link in a four bar linkage, such as a double wishbone suspension in front view, the perpendiculars to the velocity lie along the links joining the grounded link to the coupler link. This construction is used to establish the kinematic Roll center of the suspension.

See also:

* Roll center
* Screw axis