# Kinematic chain

The Watt linkage , a four-link coupling gear for guiding a rigid axle
Photo: real gear
red / blue: abstraction to the kinematic scheme
The four- link closed kinematic chain (see following picture) missing for further abstraction is the connection between the two joints fixed to the vehicle body.

Kinematic chain (English: kinematic chain) is an abstract term from transmission technology . The kinematic chain “only shows the structural relationship between the links and gives no indication of the link function. ... A gear is created from a kinematic chain if you turn one link into a frame and one or more links into a drive link. ”The kinematic scheme is the result of a first abstraction step from a real gear to a kinematic chain. It "is the structural representation of a gear that shows its essential parts for the movement, that is, limbs, joints and organs, simplified and without a structural design ..."

The representation of the chain consists of

• Symbols for the links of the gearbox (e.g. straight lines) and
• connecting small circles as symbols for the movable coupling between two links. The degree of freedom f of the joints is conveniently noted on the symbol if f> 1.

With the Watt linkage shown in the adjacent pictures - a four- link coupling gear - the abstraction from the real gear (first picture) to the four- link kinematic chain (third picture) is easy to understand. The abstract intermediate stage kinematic scheme (second picture: 1 frame , 3 moving links in real length as red straight lines and 4 joints as blue dots) can only be replaced by a square with small circles on the corners.

The abstraction to the kinematic chain always leads to an image in the plane, even if it is a complex chain, divided into loops, as it is used in chassis technology. B. can be found in the five-link suspension.

"A necessary prerequisite for a fulfilled transmission structure" is "that link groups form closed polygons", that is , that there is a closed kinematic chain . "For some ... problems .... such as industrial robots and manipulators with their grippers ... it is advantageous to consider open kinematic chains." See below.

## Further use of the term kinematic chain

In the case of robot gripper arms and similar technical devices such as B. excavator arms the term open kinematic chain is used. The same as in general transmission technology is that i. d. As a rule, two solid bodies are kinematically uniquely connected to one another to form a chain, usually with a joint with a degree of freedom f = 1. The difference is that the open chain is not inevitable. The forced running is not achieved with just one drive, but an individual drive is required in each joint. This means that a gear unit with a closed kinematic chain is added to each connection point.

The terms closed kinematic chain and open kinematic chain are also used in the biomechanics of the human body. As in transmission technology, they help with the analysis of the kinematics and the kinetics of linked links, with the majority being the individual links of the arms and legs and their movement mechanics.

## Remarks

1. Joints with f> 1 are sometimes replaced by virtual members that are connected with f = 1 joints; see. Schramm et al. a .: Modeling and simulation of the dynamics of motor vehicles .
2. The joint in the middle of the short link (the belt ) is not one of them. The Watt linkage is articulated here with the rigid axle and guides this point of the axle on the vehicle body almost on a vertical path.
3. The kinematic chain can be drawn relatively arbitrarily, but preferably as a regular polygon (in the present case as a square).

## Individual evidence

1. ^ Franz Reuleaux : Theory of Machines . Macmillan & Co, London 1876 ( limited preview in Google Book search).
2. Johannes Volmer : Transmission technology - basics . 1st edition. Technik, Berlin 1968, ISBN 3-322-93799-2 ( limited preview in the Google book search).
3. a b Kurt Luck, K.-H-Modler: Transmission technology . 1st edition. Springer, Wien - New York 1990, ISBN 3-211-82147-3 ( limited preview in Google book search).
4. Johannes Volmer, 2nd edition, 1995, page 37
5. ^ Dieter Schramm, Manfred Hiller, Roberto Bardini: Modeling and simulation of the dynamics of motor vehicles . 3. Edition. Springer Vieweg, 2018, ISBN 978-3-662-54480-8 ( limited preview in Google book search).
6. Johannes Volmer, 2nd edition, 1995, page 44
7. link.springer: "According to VDI-2861, the kinematic structure of a robot is idealized as a so-called open kinematic chain."
8. Lexicon of sports science terms: chain kinematic